My ex-girlfriend suddenly texted me to say happy birthday.

She is hinting at you that she wants to get back together with you. If the date is right, you can thank her, but she is wrong, which proves that she doesn't care so much about you. So, you can ignore her and treat her as a wrong message, or you can reply to her with thanks. It's a pity that my birthday was over yesterday, and then I ignored her, even if she sent you another message.