Is there any du`a in the Sunnah for after one has completed the Quran?

There is no specific u`a narrated in the Sunnah for after one has completed the Quran, or even from the companions of the Prophet (peace and blessings of Allah be upon him) or the well-known imams. One of the most well-known u`as regarding this matter is the u`a which is written at the end of many Mus-hafs and it attributed to Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him), but there is no basis for attributing it to him. See Fatawa al-Shaykh Ibn ‘Uthaymeen, 14/226. 

The u`a after completing the Quran is recited either after completing it whilst praying, or when not praying. There is no basis for reciting any u`a after completing it whilst praying. As for reciting a u`a after completing it when not praying, it is narrated that Anas (may Allah be pleased with him) did that. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: What is the ruling on making u`a after completing the Quran in night prayers in Ramadaan?  

He replied: 

I do not know of any u`a for completing the Quran in night prayers in Ramadaan, either from the Prophet (peace and blessings of Allah be upon him) or from his companions. The most that has been narrated concerning that is that when Anas ibn Maalik (may Allah have mercy on him) had completed a reading of the Quran, he would gather his family and make u`a. This had to do with reading it outside of prayer. End quote. 

Fataawa Arkaan al-Islam, p. 354 

Shaykh Bakr Abu Zayd wrote a useful essay on this issue, at the end of which he said: 

From the quotations in the two previous sections we may draw the following conclusion: 

1 – u`a upon completion of the Quran in general: 

The conclusion concerning this is as follows: 

Firstly: 

What we have stated above has to do with u`a upon completion of the Quran in general. 

There is no proof that the Prophet (peace and blessings of Allah be upon him) did that, rather the reports are either mawdoo’ (fabricated) or are da’eef (weak) with no other reports to strengthen them. It is almost certain that there are no marfoo’ reports concerning this matter that can be relied upon, because the scholars who wrote comprehensively about the sciences of the Quran and the dhikrs to be recited in connection with the Quran, such as al-Nawawi, Ibn Katheer, al-Qurtubi and al-Suyooti, did not narrate anything other than what we have referred to above. If they had known of any reports with stronger isnaads, they would have quoted them.

Secondly: 

There is a saheeh report that Anas ibn Maalik (may Allah be pleased with him) said u`a when he completed the Quran, and he would gather his family and children for that, and that a number of the Taabi’een followed him in that, as mentioned in the report of Mujaahid ibn Jabr (may Allah have mercy on them all). 

Thirdly: 

There is no suggestion in the texts of Imam Abu Haneefah and Imam al-Shaafa’i (may Allah have mercy on them) that this is Islamically prescribed. 

What is narrated from Imam Maalik (may Allah have mercy on him) is that this is not something that the people did, and that completing the Quran in night prayers in Ramadaan is not Sunnah. 

Fourthly: 

The view that it is mustahabb to make u`a after completing the Quran is what is narrated from Imam Ahmad (may Allah have mercy on him), as our Hanbali scholars narrated from him, and as was affirmed by some later scholars of the other three madhhabs. 

2 – u`a following completion of the Quran in prayer 

This may be summed up as follows:  

Firstly: 

From what is quoted above, there is no one letter from the Prophet (peace and blessings of Allah be upon him) or from one of his companions (may Allah be pleased with them) to indicate that it is Islamically prescribed to recite this u`a in prayer after completing the Quran, before or after bowing, whether that is done by an imam or by one who is praying alone. 

Secondly: 

The closest we can find on this matter is what was quoted by the scholars of our madhhab from the report from Imam Ahmad (may Allah have mercy on him), according to the report of Hanbal, al-Fadl and al-Harbi, the isnaad of which we cannot find, about saying the u`a for completing the Quran in Taraweeh prayer before bowing. 

According to another report narrated from him – the source of which is not known – he allowed it in the u`a of Witr. 

See Marwiyaat u`a Khatm al-Quran. 

And Allah knows best.