Allah send peace upon the birth of different Messengers. Angels brought good news from Allah of a child to different messengers also to Mother of Isa and Moses. So there is nothing wrong with celebrating birth days of messengers but we do not know the accurate dates of birth of different messengers.
1. BIRTH, DEATH AND RESURRECTION OF YAHYA
19:15 And peace be upon him the day he was born, and the day he dies, and the Day he is resurrected alive.
2. BIRTH, DEATH AND RESURRECTION OF JESUS
19:33 “And peace be upon me the day I was born, and the day I die, and the Day I am resurrected alive.”
19:34 Such was Jesus, son of Mary, and this is the truth of the matter in which they doubt.
Quran does not Prohibit Birthdays, Anniversaries, Valentine Days and other festivals as along as they do not violate the commandments of Allah.