Does the verse, “We revealed it on the night of Qadar.” [1] and other similar verses refer to the Qur’an being sent down from the al-Lawh Al-Mahfudh to the Bayt Al-Izzah or do they refer to the revelation of the first few verses of Surah Al-Alaq to Prophet Mohammed (sallallaahu alaihi wa sallam) in cave Hira?

The night of which the first verses were brought to the Prophet Muhammad (ﷺ) is not specified in any narration that I am aware of. The narrations discussing this verse all point to it referring to the revelation of the Qur’an to the Bayt Al-Izzah, as Dr Bilal Philips explains:

“Allah caused the Qur’an to descend from the Protected Tablet (al-Lawh Al-Mahfudh on which it was written) to the lowest heaven. In this revelation, all of the Qur’an was sent down at one time to a station in the lowest heaven referred to as “Bayt al-Izzah” (The House of Honor and Power). The blessed night on which this descent took place is called “Laylatul Qadar (The night of Decree), one of the odd nights in the last ten days of the month of Ramadan.” [2]

The following narration from Abdullah Ibn Abbas is found in Tafsir Ibn Kathir under the Tafsir of this verse:

“Ibn `Abbas and others have said, "Allah sent the Qur'an down all at one time from the Preserved Tablet (Al-Lawh Al-Mahfuz) to the House of Might (Baytul-`Izzah), which is in the heaven of this world. Then it came down in parts to the Messenger of Allah (ﷺ), based upon the incidents that occurred over a period of twenty-three years.'' [3]

References:

[1] Surah Al-Qadr, 97:1

[2] Dr. Bilal Philips, Usool Al-Tafseer, p. 131

[3] Tafsir Ibn Kathir, Tafsir of Surah Al-Qadr, Verse 1