I was in similar situation between two therapists except for the birthday. I think it's perfectly okay to contact her if you think it would be useful for you and I agree with Echos' points about it. I would not base my contact on her birthday though, would not even mention it, stick with therapy stuff. I actually went back to a single session with my first T after ~a year (just last week) that really validated my decision to move on from him back then. It confirmed how I viewed him/us a year ago and gave me a good feeling that I judged my situation well and made a constructive independent move.