The Quran is the scripture Muslims believe was first revealed during Ramadan.
Islamic tradition places the beginning of Quranic revelation in Ramadan, especially on Laylat al-Qadr, the “Night of Decree” or “Night of Power.” Muslims believe the angel Gabriel brought the first revelation to Muhammad while he was meditating in the Cave of Hira near Mecca. The opening verses are traditionally identified with the beginning of Surah al-Alaq.
The revelation was not delivered as a complete book in one event. Muslims believe verses and chapters were revealed gradually over approximately 23 years, from about 610 CE until Muhammad's death in 632 CE. Ramadan therefore commemorates the start of the revelation as well as continuing engagement with the completed scripture through recitation and nightly prayer.
A common confusion is the wording “first revealed.” Muslims also hold that earlier scriptures and revelations were sent to prophets such as Moses and Jesus. In this question, however, the scripture specifically associated with its first revelation during Ramadan is the Quran. The exact date of Muhammad's first revelation is debated, while Laylat al-Qadr is sought during the last ten nights of Ramadan.