Sunan Abi Dawud Book 2, Hadith 418
Abdullah ibn Ubaydullah said: I went to Ibn Abbas accompanying some youths of Banu Hashim. We said to one of them: Ask Ibn Abbas: Did the Messenger of Allah (ﷺ) recite (the Qur'an) in the noon and afternoon prayers? He replied: No. People said to him: Perhaps he might recite the Qur'an quietly. He said: May your face be scratched (a kind of curse)! This (statement) is worse than the former.
He was only a servant (of Allah) receiving Commands from Him. He preached (the divine) message which he brought with him. He did not command anything to us (Banu Hashim) specially excluding other people except three points: he commanded us to perform ablution perfectly, and not to accept charity (sadaqah) and not to make pairing of donkey with horse.
The prophet (ﷺ) prostrated himself in the noon prayer; then he stood up and bowed, and we knew that he recited Tanzil al-sajdah(surah xxxii).
Ibn ‘Isa said: No one narrated this tradition to Umayyah except Mu’tamir.
I do not know whether the Messenger of Allah (ﷺ) would recite the Qur'an at the noon and afternoon prayer or not.