Short answer
No. Allah was never born and has no birthday. He is eternal, with no beginning and no end, and the Quran says "He neither begets nor is born." Muslims have two festivals, Eid al-Fitr and Eid al-Adha, and neither marks a birth.
Believing that Allah has a beginning, a birth, parents or offspring contradicts the Quran and is kufr. Allah is al-Awwal (the First) with no beginning and al-Akhir (the Last) with no end. All Muslim schools agree.
The four schools agree on this.
Explanation
The short chapter Surah al-Ikhlas sums up what Muslims believe about Allah: He is One, the Eternal Refuge, "He neither begets nor is born, nor is there to Him any equivalent." Being born means having a beginning and depending on parents, and Allah is free of both. The Quran also calls Him "the First and the Last", and the Prophet ﷺ said that Allah existed when nothing else existed.
People usually ask this because of Christmas, which Christians keep as the birth of Jesus, whom they believe to be God the Son. Muslims believe Jesus was a human prophet, born of Mary by Allah's command. So the birth of Jesus is not, for Muslims, the birth of God. See is Jesus God in Islam?
Allah does not sleep, grow old, or change. He created time itself, so dates and anniversaries do not apply to Him. Among His names are al-Hayy (the Ever-Living) and al-Qayyum (the Self-Sustaining). See the 99 names of Allah and Surah al-Ikhlas.
The Muslim festivals are Eid al-Fitr after Ramadan and Eid al-Adha during Hajj. Both celebrate acts of worship completed for Allah, not a birthday. See Eid.
The evidence
Quran, Surah Al-Ikhlaas 112:3
لَمْ يَلِدْ وَلَمْ يُولَدْ ﴿٣﴾
He neither begets nor is born,
Allah was never born and has no offspring.
Quran, Surah Al-Hadid 57:3
هُوَ ٱلْأَوَّلُ وَٱلْءَاخِرُ وَٱلظَّٰهِرُ وَٱلْبَاطِنُ ۖ وَهُوَ بِكُلِّ شَىْءٍ عَلِيمٌ ﴿٣﴾
He is the First and the Last, the Ascendant and the Intimate, and He is, of all things, Knowing.
Allah has no beginning and no end.
Quran, Surah Al-Baqara 2:255
ٱللَّهُ لَآ إِلَٰهَ إِلَّا هُوَ ٱلْحَىُّ ٱلْقَيُّومُ ۚ لَا تَأْخُذُهُۥ سِنَةٌۭ وَلَا نَوْمٌۭ ۚ لَّهُۥ مَا فِى ٱلسَّمَٰوَٰتِ وَمَا فِى ٱلْأَرْضِ ۗ مَن ذَا ٱلَّذِى يَشْفَعُ عِندَهُۥٓ إِلَّا بِإِذْنِهِۦ ۚ يَعْلَمُ مَا بَيْنَ أَيْدِيهِمْ وَمَا خَلْفَهُمْ ۖ وَلَا يُحِيطُونَ بِشَىْءٍۢ مِّنْ عِلْمِهِۦٓ إِلَّا بِمَا شَآءَ ۚ وَسِعَ كُرْسِيُّهُ ٱلسَّمَٰوَٰتِ وَٱلْأَرْضَ ۖ وَلَا يَـُٔودُهُۥ حِفْظُهُمَا ۚ وَهُوَ ٱلْعَلِىُّ ٱلْعَظِيمُ ﴿٢٥٥﴾
Allah - there is no deity except Him, the Ever-Living, the Sustainer of [all] existence. Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them, and they encompass not a thing of His knowledge except for what He wills. His Kursi extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great.
Allah is untouched by the changes that affect created beings.
Sahih al-Bukhari 3191
Sahih... كان الله ولم يكن شيء غيره ...
"Allah existed and nothing else existed besides Him."
Allah existed before all creation, so He has no starting point.
Where the ruling is recorded
- al-Fiqh al-Akbar, the section on the attributes of Allah. States that Allah is eternal, without beginning, and resembles nothing of His creation.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.