Bananas are perfectly fine and safe in the refrigerator and no bananas do not become poisonous when they have been in the fridge.
When you buy bananas in the store those bananas have been refrigerated along their journey from where they were growing to the store you bought them in.
If the bananas would become poisonous after being refrigerated the bananas would become poisonous before they reached the store and they would not be able to sell them.
So you can safely refrigerate bananas without fear of them being poisonous.
Although when I buy bananas I usually eat them up way before I ever get a chance to refrigerate them.
I just leave the bananas out on the kitchen counter and eat them and never really needed to refrigerate them as the bananas when I buy them in the store are not refrigerated.