Can you climb mountains in your trainers?

That depends - if by 'climb' you mean a steep walk on well worn and paths with an even surface, then a sturdy pair of hiking-style trainers with good grip should suffice. Boots however are essential where good ankle support is needed; when the terrain is uneven or rocky, where it's wet or where the gradient is steep. This ankle support is especially important coming down a steep slope.

If you're actually climbing i.e. both hands and feet are required to ascend, then don't attempt it in trainers. Rock-climbers (as opposed to mountain climbers) wear very snug tight fitting light shoes specially made to allow the foot very controlled purchase on the rock face. Essential. If you're climbing the mountain rather than just specific rock routes, then boots are again hugely important and should reflect both the surface you're climbing and the weather and surface conditions (rock, snow, ice).