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.