Praise be to Allah.
Reading the Quran at the grave is not prescribed in Islam because there is no report that says that the Prophet (peace and blessings of Allah be upon him) did that.
The Standing Committee for Issuing Fatwas was asked: Is it permissible to read al-Fatihah or anything from the Quran for the deceased when visiting his grave, and does that benefit him?
“It is proven that the Prophet (peace and blessings of Allah be upon him) used to visit graves, and he would recite du`a’s for the dead that he taught to his companions and they learned them from him. For example: “Al-salamu ‘alaykum ahl ad-diyar min al-mu’minin wa’l-Muslimin, wa inna in sha Allah bikum lahiqun, nas’al Allaha lana wa lakum al-‘afiyah (Peace be upon you, O people of the dwellings, believers and Muslims. If Allah wills we will join you. We ask Allah to grant us and you safety).”
But there is no report that he (peace and blessings of Allah be upon him) recited a surah of the Quran or any verses thereof for the dead , even though he visited them often. If that had been prescribed, he would have done it and would have taught it to his companions, seeking the reward and out of mercy towards his Ummah, and fulfilling the obligation to convey the message. For he was as Allah described him (interpretation of the meaning):
“Verily, there has come unto you a Messenger (Muhammad) from amongst yourselves. It grieves him that you should receive any injury or difficulty. He (Muhammad) is anxious over you; for the believers (he is) full of pity, kind, and merciful.” [At-Tawbah 9:128]
The fact that he did not do that even though there were reasons why he could have done it indicates that it is not prescribed. His companions (may Allah be pleased with them) knew that and followed in his footsteps, and they limited themselves to learning lessons and making Du`a for the dead when they visited them. There is no report that they read the Quran for the dead . For them such reading was an innovation (Bid`ah), and it was proven that the Prophet (peace and blessings of Allah be upon him) said: “Whoever introduces anything into this matter of ours [i.e., Islam] that is not part of it will have it rejected.” (Narrated by Al-Bukhari and Muslim) (Fatawa Al-Lajnah Ad-Da’imah, 9/38)
And Allah knows best.