Allahu ta ‘ala especially mentions the births of several Prophets as a mercy, this establishes that these are auspicious occasions:
Sayyidinaa Yahya ‘alaihi salaam
This is the remembrance of the mercy of your Lord upon His bondman Zakaria. When he softly prayed to his Lord. [Surah Maryam 19:2-3]
Note that Allah has called the birth of Sayyidinaa Yahya a “mercy”, and it’s mentioning in the Noble Qur’an as a “remembrance”. What greater evidence would be required for the remembrance of the Beloved Prophet’s birth?
Sayyidinaa ‘Isa bin Maryam ‘alaihi wa ‘alaiha salaam
“And peace is upon me the day I was born, and on the day I shall taste death, and on the day I will be raised alive.” This is ‘Isa (Jesus), the son of Maryam; a true statement, in which they doubt. [Surah Maryam 19:33-34]
Allah mentions the birth of several prophets at the same time:
And when Musa said to his people, “O my people! Remember Allah’s favor upon you, that He created Prophets among you, and made you kings, and has now given you what He has not given to anyone else in this world.” [Surah Maidah 5:20]
Note that Allah has mentioned the creation of Prophets among the Bani Israel, separately from other favors, in order to especially highlight this.