Is it true that one of the Salaf (pious predecessors) used to complete a full recitation of the Qur’an daily for a month as a dowry for a houri, and is it possible to do that to obtain a houri, and have any hadiths been narrated concerning this?

The predecessors (Salaf) did not transmit anything mentioning a dowry for the Houris (maidens of Paradise). However, Al-Qurtubi mentioned accounts of dreams in which some of the predecessors saw the dowries of the Houris, including the recitation of the Qur’an. Among these, Muhammad ibn al-Nu'man al-Maqri mentioned that al-Jala al-Maqri informed him about a Sheikh who "bought" a Houra from Allah with four thousand recitations (of the Qur’an). When he completed them, he saw her in a dream, and she said she was the Houra he had "bought," and she asked him for another thousand recitations as a dowry.

Mudarr al-Qari mentioned that he overslept and missed his regular portion (of Qur’an recitation) and saw a maiden who gave him a parchment with verses urging him to recite the Qur’an and abandon sleep.

Similarly, Malik ibn Dinar mentioned that he overslept and missed his regular portions (of Qur’an recitation) and saw a maiden who gave him a parchment with verses urging him to perform night prayers (Tahajjud) with the Qur’an and abandon sleep.

There are no Hadith (definition) that indicate that whoever completes the recitation of the Qur’an will have a Houra. However, the intention behind completing the Qur’an must be to worship Allah and attain His reward, which includes entering Paradise, and among its blessings are the Houris.