From what I've seen of the storylines, Quinn is only pissed at Josy for cracking a "short" joke when they very first meet - from there on, her attempts to stop the duo joining are aimed at Maya. Now obviously if Quinn succeeds in keeping Maya out of the sorority, Josy doesn't get in either; but I don't see that Quinn particularly doesn't want Josy to join, it's Maya she is trying to block. (It could in fact be both, but I see no evidence that it is...)for me they are almost synonymous... antagonist is more like narrative, nemesis is more like comics or bible
if she could do it without Quinn, she could do it now after the scavenger hunt, what would prevent it now? or right away, since Josy didn't seem to believe in free tuition (assuming that with the money for a membership and lots of love you can pay for 2 memberships...)
for you Quinn with his actions has damaged Josy, Josy of her own has not realized and does not even show a particular bitterness towards him (which for example has against Tommy)
and all this has nothing to do with the fact that Quinn remains a negative character, with many shadows and very little light.
On the other hand, whether Quinn's screwing over Josy deliberately or just as collateral damage, doesn't affect Josy being screwed over ...