No, not really. People just say: it will be cloudy, which was your original sentence. If you really want to emphasise that there will be a lot of individual clouds, I suppose you can say: there will be lots of clouds. The only situation I can imagine someone would say that is if two people wish to go cloud watching, and person A says: there won't be any clouds to see, and person B disagrees, and says: there will be lots of clouds. Otherwise, in the general sense of weather: it will be cloudy.

Konnichiwa We pretty much just say (it is cloudy) or it will be cloudy would work pending on if there will be a storm moving in.