Is it correct to use "congratulate" for someone's birthday?

As a native speaker, I can tell you that "congratulations" is more often used on occasions such as weddings, engagements, graduations, job promotions, and births (or expectant); but not so much for birthdays and other holidays, though it is not so much a matter of correctness here as it is custom. If you wanted to, though, you could correctly say:

"I'm not going to congratulate her on her birthday." or

"I'm not going to wish her a happy birthday."

For occasions such as holidays, it is most natural to say, for example:

"Happy Mother's/Veteran's/Teacher's/Star Wars Day" rather than using "Congratulations" for such occasions.