Did I really believe she would agree?
Did I really believe she would have agreed?
What's the difference between the two? Is one more common/grammatically correct than the other?
They convey slightly different meanings:
1) Did I really believe (then that) she would agree (with me now [or more recently])?
2) Did I really believe (then) that she would have agreed (with me back then)?
Sometimes you hear the first one used like the second. This is probably a result of people getting lazy in conversation and saying "Did I really believe that she would agree?" instead of "Did I really believe that she would've agreed?" The word "would've" is really cumbersome to pronounce so just leaving the (have) and (-ed) implicit is more instinctive in casual conversation if there is context to support the story.
OP's second version is cumbersome, ambiguous, and probably wouldn't be used at all. Belief and agreement are slippery concepts when we've got complex tense interactions like this, so I'll recast to something I find easier to deal with...
or more simply, since the fact of it being framed as a question doesn't affect the "awkward" part...
It's then much easier to see that in #3 what I knew (and what OP wondered if he believed) was that she would go at some later time (after the knowing/wondering).
But what I knew in #4 could be either of two possibilities (both grammatically credible)...
It's a bit more tricky to transfer the grammar of 4a/b to the semantics of OP's two versions.
To make sense of the 4a interpretation we have to assume her (already-made) agreement was with someone else (I could hardly be wondering whether she'd already agreed with me).
A simple "reason" that works for the 4b interpretation is to suppose that I never in fact asked her, which is why she didn't actually agree. But of course there could be any number of other reasons.
Regarding which form is more common, here are some relevant counts from Google Books,...