Most Helpful Girl
Well i'm assuming y'all don't even talk anymore, not friends either.
It's up to you but know that she may find it annoying, or really like it. Depends on what terms you guys ended.
Most Helpful Guy
Depends on how did the breakup go. Initiating conversation could start things up again or it could end with her not replying and you questioning why you did it in the first place. A birthday wish might give her the impression that you care for her (which even if you do, its wrong in this period of time for a breakup)