Not within a week, but within a few weeks it is.

It is true the movement of plates is very slow, but flooding can happen remarkably fast. if you have lowlands suddenly joined to the ocean a few weeks to fill is possible. The mediterranean (Zanclean) flood is a good example as is the bonneville flood. At one time you could walk from Italy to Sicily to Tunisia (africa), within a few weeks there was a sea in the way. When gibraltar breached the ocean flooded into the the lowland, some estimates put the flooding at over 100,000,000 cubic metres per second. Erosion turned a tiny breach into a huge gaping chasm in minutes. The flooding was also incredibly noisy I remember one claim that noise might have deafened local wildlife. Afterwards there was a sea where there had been crossable land before.

In my opinion, it would be impossible for a continent to split apart and the two parts move far enough away to separate two warring tribes, nations, or civilizations within a short time, a time so short that the war would not have ended one way or another long before the separation process was completed.

One way to get around that would be to have two separate continents that were connected by a land bridge. There is a chain of islands between them and during a glacial period the islands are all covered with ice sheets but the sea level is so low that enough of the seabed between the islands is both exposed and ice free that a land path winds between the glaciers. The land path might be narrow in some places but wide enough in other places for lush and bountiful lowlands that might be the best lands on the planet at the moment, well worth fighting over.

So during a interglacial period seawater flows between the separate islands from one ocean to another and the waters of those oceans mix and tend to have similar temperatures. So the colder ocean stays warm enough to stay ice free and the dark water adsorb more solar heat than ice would, thus keeping that ocean warmer still, and reinforcing the interglacial period.

But during a glacial period seawater cannot flow through the land bridge from one ocean to another and the waters of one of those oceans gets colder and colder and it fills up with pack ice, lowering temperatures worldwide and reinforcing the glacial period.

During a glacial period the two continents will seem like one continent to land dwellers even if technically the two continents are separated by hundreds or thousands of kilometers or miles. The two different cultures will expand in opposite directions through the land bridge until they meet and fight over control of one or more of the large lush lowlands on the exposed seabed - which of course they don't know is exposed seabed.

And possibly both cultures are landlubbers who don't have any experience sailing the seas.

Then a vast volcanic eruption or an asteroid impact destroys a narrow section of the land bridge. The after effects might possibly trigger an even stronger glaciation, so that the entire area of the land bridge becomes buried under an ice sheet and impassable for centuries or millennia.

Or the after effects might possibly trigger some degree of global warming. Ice sheet may melt and raise sea levels. And with a part of the lowlands devastated and at a lower level, water can now pass through the former land bridge from the warmer ocean to the cooler ocean, raising the water temperature in the cooler ocean and melting ice on and around that ocean, raising sea levels and adsorbing more solar heat to melt more ice and raise sea levels even more, and so on.

By the time the after effects are over and people of the two cultures on the two continents begin to return to the devastated lands of the land bridge, the entire land bridge might be flooded by the rising sea levels, and possibly the nearest islands in the island chain will be out of sight of land. Then the two cultures on the two continents will be separated until one of them develops ships and learns how to sail between continents.

Another possibility would be the opposite, having two cultures that live on land, but have fishing and seaborne trade as important parts of their economy, so never explore or settle more that a few dozen miles from the ocean.

And possibly two different oceans that the two different cultures live on the shores and islands of are separated by two great land masses with only a winding, narrow, strait connecting those oceans. The climate may be bleak and inhospitable at the strait and near it, so nobody settles there to live, but powerful societies might build forts and station fleets there, and supply them with food and water, etc. from elsewhere, to control the vital trade from one ocean to another. So that would explain the war between societies in the two oceans.

Then a volcano erupts in the strait and pours out many cubic miles of lava that solidifies and block the entire strait. Then it will be impossible from people from the two oceans to ever meet again, until someone finds another passage between the oceans through even less hospitable land, or another volcanic event blows away the plug in the strait, or one of the cultures begins to explore and travel long distances on land.

Or possibly there is an asteroid strike somewhere in the planet, that causes a glaciation, with icecaps forming and sea levels falling. All the port cities will have to move closer to the receding shores as the sea levels drop. And possibly the strait will become dry land as sea levels fall. and if the continental shelf around the strait is flat enough, the exposed land blocking the strait could be hundreds or thousands of kilometers or miles wide.

A meteor strike.

Meteor Crater formed in a fraction of a second as 175 million tons of
limestone and bedrock were uplifted, forming the mile-wide crater rim
in the formerly flat terrain. The meteorite was only 150 ft. wide. For
a sense of scale, if this hit Kansas City, the blast radius would take
out the entire city. (ref)

Now, in this example, Barringer Meteor Crater in Arizona USA, the crater is not quite a mile wide (0.737 miles, 1.186 km), so it's very easy to traverse. But not at first.

Imagine this happens in an area with forests or other vegetation that catches on fire. Imagine that settlements in the middle of this large community were wiped out in an instant. And remember that the blast radius here is huge.

Using scaling relationships determined from nuclear explosions, the
radial extent of the air blast produced by the Meteor Crater impact
event is estimated. The wind velocity at a distance of 5 crater radii
(3 km) from the point of impact should have exceeded 2000 km/h.
Hurricane force winds would have existed as far away as 20 to 40 km,
depending on the exact explosive energy of the impact event. To
determine how this event may have affected the environment surrounding
the crater, the topography, vegetation, and animal life that existed
at the time of the impact are reconstructed. For example, if the
coniferous woodlands were 100 m lower than they are presently and they
had moved farther out onto the plains, then the air blast would have
flattened trees within a 16 to 22 km radius of the point of impact and
damaged them over an area of 4100 to 8500 km2. The distance over which
the damage occurred may have been up to 2× larger in some directions
around the crater because of additional effects produced by the
ballistic shock wave. (ref)

So sure, it's a simple trip now, but a blast that flattens trees up to 22 kilometers (13.5 miles) away, would easily be enough to separate a community for generations.

The stories they must have! And the fear of going anywhere near it. How could they know it wouldn't happen again? Maybe they'd know the impact came from the sky (if it was during the day and if the stories remember that bit), maybe not. Were the Gods angry at them for building villages in that location? Or is there some evil lurking under the surface ready to destroy any fools who venture too near?

If the cultures on either side of this were already at war, I think this would seal the deal. War over. Okay, let's go that way from now on.

If you want water separating them, a meteor crater could could land in such a way to turn a passible river into an impassible one. Or at least one that's more trouble than it's worth. In 10-20 years when the trees have grown back and people get curious, the river could have swelled. But you don't really need a water barrier to have a psychological one. Who wants to go near that place again?

Add in some geographical features like a mountain range or a waterless desert if you want to make going around even harder. Set it up however you wish.

Creating a storyline with a natural phenomenon dividing people in less than a week, lasting for generations, is quite doable. With the right geography, politics, technological level, and so forth, it's very reasonable.

With a long, deep valley I think it's entirely possible for erosion to gradually chip away at one end, until there's the equivalent of a dam breaking, and a river changes course to flood the valley and probably even increasing the size of the rift with debris being swept away and sudden erosion taking out the banks. Depending on geography it could be the ocean flooding in as well, I suppose. You could feasibly end up with something like the Grand Canyon, which could be possible to cross with enough work (bridges, climbing down and rafting over, etc,) but it would take a lot of work and risk, so unless there's something like a major food source on the other side people probably wouldn't bother.

Depending on how long it is and where the other end is it will probably still be possible to walk around, but it might take weeks or months, and as with going over that's a big commitment of time and resources.

Megaflood from melting ice dam.

When glaciers melt, huge amounts of water can pile up behind ice dams. When those dams give way the water comes out. Fast. Supposedly the Hudson River valley was carved in a couple of weeks. I am glad I was not there at the time.

https://io9.gizmodo.com/ancient-flood-myths-may-have-a-basis-in-geological-hist-5940112

There is now compelling evidence for many gigantic ancient floods
where glacial ice dams failed time and again: At the end of the last
glaciation, some 10,000 years ago, giant ice-dammed lakes in Eurasia
and North America repeatedly produced huge floods. In Siberia, rivers
spilled over drainage divides and changed their courses. England's
fate as an island was sealed by erosion from glacial floods that
carved the English Channel. These were not global deluges as described
in the Genesis story of Noah, but were more focused catastrophic
floods taking place throughout the world. They likely inspired stories
like Noah's in many cultures, passed down through generations.

Your continent had such a flood. A huge amount of water rushed thru low points, wearing them away. Once it was done, the land may have been eroded low enough for the ocean to come up a fair ways along the course of the flood. What was one landmass has become two.

Yeah, I took my own answer from here
How might a naturally-occuring geographical barrier between two areas *suddenly* become permeable so that a few people can cross it?

But the questions are not really duplicates I don't think.

Earth

No. Continental drift moves at breakneck speed of about 2.5cm per year. Some places even faster. Even major earthquakes haven't been able to produce more than relatively tiny cracks.

And there are fits and starts, notably along the Great Rift Valley. Perhaps in ten million years, warring cultures in this region of Africa will become separated by a sea.

AltEarth

Maybe. Catastrophic continental drift is apparently a thing on some Earths and is used to support other claims. You probably don't want them quite that catastrophically separated by water!

A Certain Fictional World

There was a sea that had been recently cut off from the Ocean and over the course of years was evaporating at a pace faster than rivers and precipitation can replenish. A ruler of an affected country seeks for a solution. Digging a canal through the blockage is too much like work and would be insufficient to establish flow, so the obvious alternative is to turn to Magic! Massive thoumically sounded sub-sonic horns are constructed along the desired route of the split, the sound of which is intended to "tickle the ears" of those Great Beings that hold up the continents. They got tickled alright. The Beings were so annoyed with the sound, transmitted through miles of rock, that they began to shift about most uncomfortably. Earth tremors were the prevailing result over an area of many hundreds of square miles. In a more specific location, an existing rift opened up causing the creation of a new strait. In the end everyone was happy: the earthquakes destroyed all the nuisance horns; the dying sea refilled; all the countries around gained a new and exciting rationale to wage war on each other

Before the event we can see deep in the earth a fissure, and the underlying Forces are content:

However, after the event, we can see that the fissure opened up wide and an emergency scaffold has been placed upon a mantle plume to support the cooling plug that underlies the new Strait. Also, the underlying forces are Not Amused by the abuse to their senses: