Praise be to Allah.
Firstly:
The Messenger (peace and blessings of Allah be upon him) forbade reciting Quran when bowing and prostrating.
Muslim (479) narrated that Ibn ‘Abbaas (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “I have been forbidden to recite the Quran when bowing or prostrating. As for bowing, glorify your Lord therein, and as for prostrating, strive in du`a, for it is deserving of a response (from your Lord).”
Muslim (480) narrated that ‘Ali ibn Abi Taalib (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) forbade me to recite Quran when bowing and prostrating.
The scholars are agreed that it is makrooh to recite Quran when bowing or prostrating.
See al-Majmoo’, 3/411; al-Mughni, 2/181
The wisdom behind that is:
It was said that the best pillar of prayer is standing and the best of dhikr is reciting Quran, so the best has been put with the best, and it is not allowed to put it with anything else, lest anyone think that it is equal to other kinds of dhikr. ‘Awn al-Ma’bood.
It was said that it is because the Quran is the noblest of speech, because it is the Word of Allah, and the position of bowing and prostration is one of humility and submission on the part of the worshipper, so it is more appropriate not to recite the Word of Allah in these two positions. Majmoo’ al-Fataawa, 5/338
Secondly:
If a person recites a du`a that is mentioned in the Quran when prostrating, such as the words (interpretation of the meaning): “Our Lord! Give us in this world that which is good and in the Hereafter that which is good, and save us from the torment of the Fire” [al-Baqarah 2:201], there is nothing wrong with that, if the intention is to say du`a and not to recite Quran, because the Prophet (peace and blessings of Allah be upon him) said: “Actions are but by intentions, and each person will have but that which he intended.” Narrated by al-Bukhaari, 1; Muslim, 1907.
Al-Zarkashi said: It is makrooh when it is intended as recitation of Quran, but if it is intended as du`a and praise, then it is like someone who did Qunoot by reciting a verse. End quote.
Doing Qunoot by reciting a verse from the Quran is permissible and is not makrooh.
Tuhfat al-Muhtaaj, 2/61
Al-Nawawi said in al-Adhkaar (p. 59)
If a person does Qunoot by reciting a verse which includes a du`a, then he has done Qunoot, but it is better to say the du`a as narrated in the Sunnah. End quote.
This applies if he intends to say du`a when he recites the verse.
See al-Futoohaat al-Rabbaaniyyah Sharh al-Adhkaar al-Nawawiyyah by Ibn ‘Allaan, 2/308
The scholars of the Standing Committee were asked: We know that it is not permissible to recite Quran when prostrating, but there are some verses which include du`as, such as the words, “Our Lord! Let not our hearts deviate (from the truth) after You have guided us” [Aal ‘Imraan 3:8]. What is the ruling on saying such du`as that are mentioned in the Quran, when prostrating?
They replied: There is nothing wrong with that if it is done as a du`a and not as recitation of Quran. End quote.
Fataawa al-Lajnah al-Daa’imah, 6/443.
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