We all know about Rain Shadow - And if you don't now you do - and that the inner side of a mountain as compared to the ocean would be drier.

Now a worldbuiding problem going around that phenomenon. Imagine a large and wide valley (several dozens of miles wide in all directions) surrounded by tall mountains in all sides. Should be a desert, right? Except it is not. It is a lush and fertile land, with rainy seasons and extensive flora and fauna.

How can it be possible?

This is not reality-check so no frame challenges please. Stay science-based so no magic. Within the premises above, what conditions would cause such an environment?

Simplest answer is a very large warm water lake in the centre (Around the size of the great lakes if you can, but even a lake half as big will work, even as little as 25km in diameter is good enough to have some effect). The lake will change the pressure in the area, creating some small amounts of rain. The water table can be close to the surface, allowing trees to grow, and there can be frequent flooding of this lake (makes sense, the water has no where to go when it rains).

And that gives you a realistic fertile land that rains sometimes.

Here is a link explaining some of the effects the great lakes have on the weather.
http://greatlakesliteracy.net/principles/3/

The Great Lakes influence local and regional weather and climate.

A:
The Great Lakes affect weather and climate by impacting the basin’s energy and water cycles. Changes in the Great Lakes’ water
circulation, water temperatures and ice cover can produce changes in
weather patterns.

B:
The Great Lakes warm by absorbing solar radiation. Lake temperatures are also affected locally by the temperature of inflowing
river waters. The Great Lakes lose heat by evaporation and by warming
the overlying air when the atmosphere is cool. After water vapor is
released into the atmosphere, it condenses and forms precipitation,
some of which falls within the Great Lakes Basin.

C:
The Great Lakes modify the local weather and climate. Because water temperatures change more slowly than land temperatures, lake
waters gain heat in summer and release heat during cooler months. This
results in cooler springs, warmer falls, delayed frosts and lake
effect snow.

D:
The Great Lakes have a significant influence on regional climate by absorbing, storing and moving heat and water. Lake effect
precipitation can occur downwind when major weather systems move over
the lakes.

E:
The Great Lakes are influenced by larger climate change patterns affecting North America and the world. Climate patterns in the Great
Lakes are changing, with warmer and drier conditions predicted.

Depending on the height of the mountains, you have an interesting secondary concern, shorter days. This will help keep the area cool during the day, allowing for people to work harder and allowing plants to grow without being dried out.

Because the mountains aren't high enough to block all the weather systems that come that way. This can be a function of:

A. Them actually being relatively low all around i.e. less than 1000m above sea level so they just don't block the weather, mountains of this size may seem small but if they're extensive and broken enough they'd still be impassable.

or

B. The mountains have wide passes at relatively low altitude that lets weather systems into the valley when they're moving to the right direction, this could make for a distinct wet and dry season if there is something like a monsoon weather cycle operating.

Alternatively the water may be in the form of glacial melt in which case the precipitation that supplies in need not occur within the valley but rather on outward facing slopes with the glaciers twisting back into the valley. Or even in the form of springs feeding through porous rock into the valley from outside.

The fertility is a given, if there's enough water coming into the valley it will be depositing rich sediment into the valley bottom creating and renewing a rich soil layer.

I give you Kelowna, British Columbia, and the Okanagan Valley. Surrounded by mountains, it's a humid continental climate with no shortage of vegetation and is certainly no desert.

For a larger example, the entire interior of British Columbia: between the Coast Range and the Rockies, the Interior consists of topographically lower plateaus and valleys with assorted climates, none of them desert-like.

There are a number of options.

Glaciers

There are glacier vallies up in the mountains and ithe ice flows provide water.

Aquifier surfaces

There is a water rich layer of rock, an underground river providing water to the valley for the ocean side.