Short answer
The Quran began to be revealed in Ramadan, on Laylat al-Qadr, around 610 CE, when the Prophet Muhammad ﷺ was about 40 and in the cave of Hira near Makkah. It came down in parts over about 23 years and was complete shortly before his death in 632 CE.
This is a matter of history and belief, not a fiqh ruling. All Muslims believe the Quran is the uncreated speech of Allah, revealed in stages through the angel Jibril.
The four schools agree on this.
Explanation
The first revelation. The Prophet ﷺ used to go to the cave of Hira on the mountain of Light outside Makkah to worship alone. There the angel Jibril came to him and said Iqra, "Recite", and taught him the first ayat of Surah al-Alaq (96:1 to 5). Aisha's account of this is in the opening pages of Sahih al-Bukhari. The Quran itself says it was sent down in the month of Ramadan (2:185) and on the Night of Decree (97:1).
Revealed in stages. The Quran did not come down as one book in one day. It came over about 23 years, 13 in Makkah and 10 in Madinah, in passages that answered events and questions as they happened (17:106). The scholars explain, from a report of Ibn Abbas, that it was first sent down as a whole to the lowest heaven on Laylat al-Qadr, then revealed to the Prophet ﷺ bit by bit. Each Ramadan, Jibril reviewed with him all that had been revealed.
Written and collected. Scribes wrote the ayat as they came, and many Companions memorised them. Abu Bakr had the whole Quran gathered into one copy after the battle of Yamamah, and Uthman had standard copies made and sent to the main cities. The structure and history are told in the history of the Quran.
The dates in years are approximate, since the Arabs then did not use a fixed calendar. What is sure is the month, Ramadan, which is why recitation in Ramadan carries special weight: see Laylatul Qadr.
The evidence
Quran, Surah Al-Baqara 2:185
شَهْرُ رَمَضَانَ ٱلَّذِىٓ أُنزِلَ فِيهِ ٱلْقُرْءَانُ هُدًۭى لِّلنَّاسِ وَبَيِّنَٰتٍۢ مِّنَ ٱلْهُدَىٰ وَٱلْفُرْقَانِ ۚ فَمَن شَهِدَ مِنكُمُ ٱلشَّهْرَ فَلْيَصُمْهُ ۖ وَمَن كَانَ مَرِيضًا أَوْ عَلَىٰ سَفَرٍۢ فَعِدَّةٌۭ مِّنْ أَيَّامٍ أُخَرَ ۗ يُرِيدُ ٱللَّهُ بِكُمُ ٱلْيُسْرَ وَلَا يُرِيدُ بِكُمُ ٱلْعُسْرَ وَلِتُكْمِلُوا۟ ٱلْعِدَّةَ وَلِتُكَبِّرُوا۟ ٱللَّهَ عَلَىٰ مَا هَدَىٰكُمْ وَلَعَلَّكُمْ تَشْكُرُونَ ﴿١٨٥﴾
The month of Ramadhan [is that] in which was revealed the Qur'an, a guidance for the people and clear proofs of guidance and criterion. So whoever sights [the new moon of] the month, let him fast it; and whoever is ill or on a journey - then an equal number of other days. Allah intends for you ease and does not intend for you hardship and [wants] for you to complete the period and to glorify Allah for that [to] which He has guided you; and perhaps you will be grateful.
The Quran was first sent down in Ramadan.
Quran, Surah Al-Qadr 97:1
بِّسْمِ ٱللَّهِ ٱلرَّحْمَٰنِ ٱلرَّحِيمِ إِنَّآ أَنزَلْنَٰهُ فِى لَيْلَةِ ٱلْقَدْرِ ﴿١﴾
Indeed, We sent the Qur'an down during the Night of Decree.
Its revelation began on Laylat al-Qadr.
Quran, Surah Al-Israa 17:106
وَقُرْءَانًۭا فَرَقْنَٰهُ لِتَقْرَأَهُۥ عَلَى ٱلنَّاسِ عَلَىٰ مُكْثٍۢ وَنَزَّلْنَٰهُ تَنزِيلًۭا ﴿١٠٦﴾
And [it is] a Qur'an which We have separated [by intervals] that you might recite it to the people over a prolonged period. And We have sent it down progressively.
It was revealed in parts over many years.
Sahih al-Bukhari 6
Sahih... وكان يلقاه في كل ليلة من رمضان فيدارسه القرآن ...
"Jibril used to meet him every night of Ramadan and go over the Quran with him."
The Quran was reviewed with the Prophet every Ramadan as it was being revealed.
Sahih al-Bukhari 4986
Sahih... فتتبعت القرآن أجمعه من العسب واللخاف وصدور الرجال ...
"Zayd ibn Thabit said: So I searched for the Quran and gathered it from palm stalks, thin stones and the memories of men."
Describes the first collection of the Quran into one copy under Abu Bakr.
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.