I've always thought FR/LG was Gen I because it was in the Kanto region, and the same for HG/SS but I thought it was Gen II.
I think I'm wrong now, but I'm really not sure.
Yes they are. FR/LG is considered 3rd gen and HG/SS is considered 4th gen. This is because you can get Pokemon from 2nd and 3rd gen in FR/LG and you can get Pokemon from 3rd and 4th gen in HG/SS. Also, in FR/LG there are 3rd gen mechanics such as abilitites and natures as well as items and other 3rd gen features and HG/SS has 4th gen mechanics, abilities, etc.
(My source also marks FR/LG as 3rd gen and HG/SS as 4th gen)
Simply put, yes.