you could say that. i guess. but if you want to sound more native I (personally) would say something like: I got sick a lot this month because of the bad weather. or to shorten it a bit you could say "I got sick a lot this month"
saying "fell ill" is almost never said unless you want to act old and stuffy, hahah. You would hear that in a novel, book, or play that was set decades ago. you can say "ill" if you want, but again, its uncommon.