Short answer... no they aren't dangerous, in fact bananas are a great source vitamins and minerals, and especially in the pregnancy months where you may not be getting enough. It is especially high in potassium which is great for (just watch out as they can cause constipation).

As for the watermelons... well that is a Chinese belief that during pregnancy you should avoid cooling foods as they make your Qi weaker. I think its the Qi. Anyway, theres a whole lot of thought behind it, but not a lot is valid.