In The Fellowship of the Ring, Gloin and Gimli attend the Council of Elrond to discus the One Ring. Legolas is also in attendance. In the movies at least, Legolas is a major part of the story and is a major factor in allowing the Dwarves to reach Erebor. Plus it's only about 60 years between the end of the events of the Hobbit and the creation of the fellowship of the ring and those events aren't something either a dwarf or an elf would forget very easily. So, my question is actually a two parter. Why is Gloin so hostile towards Legolas and the elves during the Council of Elrond? And why do the two not even acknowledge each other? I don't remember exactly, but I'm fairly certain that Legolas wasn't a part of the events of the Hobbit in the book. Should I just chaulk this up to movie inconsistency?
Movie inconsistency. Legolas was not written into the Hobbit.
If you want a canon answer, they only interact once, and Legolas is an elf and also insults his son, might make you ignore him.
It's part of the nature of Tolkien's Dwarves that they are slow to forget grievances; for example from the Silmarillion (Of Aule and Yavanna):
Therefore they are stone-hard, stubborn, fast in friendship and in enmity...
Legolas and Glóin actually do have some (brief) interaction in the books, at the Council of Elrond, where Glóin shows in no uncertain terms that he still bears some resentment over his imprisonment:
'You were less tender to me,' said Glóin with a flash of his eyes as old memories were stirred of his imprisonment in the deep places of the Elven-king's halls.
As you said, "those events aren't something either a dwarf or an elf would forget very easily", and this Dwarf certainly didn't.