The Messenger of Allah ﷺ used to recite in the noon and afternoon prayer: "By the Heaven and the Morning Star" (Surah 86) and "By the Heaven , holding mansions of the stars" (Surah 85) and similar surahs of equal length.  

أَنَّ رَسُولَ اللَّهِ ﷺ كَانَ يَقْرَأُ فِي الظُّهْرِ وَالْعَصْرِ بِالسَّمَاءِ وَالطَّارِقِ وَالسَّمَاءِ ذَاتِ الْبُرُوجِ وَنَحْوِهِمَا مِنَ السُّوَرِ  

When the sun declined, the Messenger of Allah ﷺ offered the noon prayer and recited surahs lie "By the night when it covers over" (92) and (recited similar surahs) in the afternoon prayer, and in the other prayers except the dawn prayer which he used to prolong.  

كَانَ رَسُولُ اللَّهِ ﷺ إِذَا دَحَضَتِ الشَّمْسُ صَلَّى الظُّهْرَ وَقَرَأَ بِنَحْوِ مِنْ { وَاللَّيْلِ إِذَا يَغْشَى } وَالْعَصْرَ كَذَلِكَ وَالصَّلَوَاتِ كَذَلِكَ إِلاَّ الصُّبْحَ فَإِنَّهُ كَانَ يُطِيلُهَا  

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.  

أَنَّ النَّبِيَّ ﷺ سَجَدَ فِي صَلاَةِ الظُّهْرِ ثُمَّ قَامَ فَرَكَعَ فَرَأَيْنَا أَنَّهُ قَرَأَ تَنْزِيلَ السَّجْدَةِ قَالَ ابْنُ عِيسَى لَمْ يَذْكُرْ أُمَيَّةَ أَحَدٌ إِلاَّ مُعْتَمِرٌ  

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.  

دَخَلْتُ عَلَى ابْنِ عَبَّاسٍ فِي شَبَابٍ مِنْ بَنِي هَاشِمٍ فَقُلْنَا لِشَابٍّ مِنَّا سَلِ ابْنَ عَبَّاسٍ أَكَانَ رَسُولُ اللَّهِ ﷺ يَقْرَأُ فِي الظُّهْرِ وَالْعَصْرِ فَقَالَ لاَ لاَ فَقِيلَ لَهُ فَلَعَلَّهُ كَانَ يَقْرَأُ فِي نَفْسِهِ فَقَالَ خَمْشًا هَذِهِ شَرٌّ مِنَ الأُولَى كَانَ عَبْدًا مَأْمُورًا بَلَّغَ مَا أُرْسِلَ بِهِ وَمَا اخْتَصَّنَا دُونَ النَّاسِ بِشَىْءٍ إِلاَّ بِثَلاَثِ خِصَالٍ أَمَرَنَا أَنْ نُسْبِغَ الْوُضُوءَ وَأَنْ لاَ نَأْكُلَ الصَّدَقَةَ وَأَنْ لاَ نُنْزِيَ الْحِمَارَ عَلَى الْفَرَسِ  

I do not know whether the Messenger of Allah ﷺ would recite the Qur'an at the noon and afternoon prayer or not.  

لاَ أَدْرِي أَكَانَ رَسُولُ اللَّهِ ﷺ يَقْرَأُ فِي الظُّهْرِ وَالْعَصْرِ أَمْ لاَ