There is nothing wrong with awarding and receiving prizes in competitions for memorising the Holy Qur’an, whether these prices are awarded by a third party, other than the contestants, or by one of the contestants or all of them, according to the more correct view, because that is akin to what the Lawgiver mentioned of it being permissible to compete for prizes in archery, camel racing and horseracing, because that comes under the heading of supporting Islam and making it supreme. Abu Dawood (2574), at-Tirmidhi (1700) and Ibn Maajah (2878) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “There should be no prizes except in archery, camel-racing or horse-racing.”

Classed as saheeh by al-Albaani in Saheeh Abi Dawood 

Ibn al-Qayyim (may Allah have mercy on him) said: 

Is it permissible to compete for prizes in memorising Qur’an, hadeeth, fiqh and other beneficial sciences, and getting questions right? The companions of Maalik, Ahmad and ash-Shaafa‘i disallowed it, but the companions of Abu Haneefah and our shaykh regarded it as permissible; Ibn ‘Abd al-Barr also narrated that from ash-Shaafa‘i, and prizes are more appropriate in wrestling and swimming. If it is regarded as permissible to compete in these fields for prizes, then it is more appropriate that it should be permissible to compete in knowledge. This is similar to the case when (Abu Bakr) as-Siddeeq made a bet with the disbelievers of Quraysh regarding the truth of what he told them. We have stated previously that there is no shar‘i evidence to suggest that this has been abrogated, and Abu Bakr took the wager from them after gambling had been prohibited. Religion is established by means of proof and jihad; if betting was permissible with regard to tools of jihad, then it is more appropriate that it should be permissible with regard to knowledge. This view is the correct one.

End quote from al-Faroosiyyah (p. 318) 

See also: question no. 138652 

This has to do with memorising the Qur’an or memorising the Sunnah, not just reading and completing. Memorising takes effort, and increasing the numbers of those who have memorised the Qur’an, and raising high the banner of the Qur’an, is one of the greatest kinds of beneficial knowledge on which the religion is based. 

With regard to offering prizes for merely completing a reading of the Qur’an, there may be some reservations about it. The more correct view is that it is not prescribed, because it involves making known actions which should be concealed, boasting about that, and causing the heart to be attached – with regard to an act of worship – to these prizes for which people are competing, which then leads to being hasty in reading in order to complete the reading. Holding competitions for merely completing the Qur’an is something about which there may be reservations, so how about if prizes are awarded for that, taken from the contestants’ money? 

Shaykh ‘Abd ar-Rahmaan al-Barraak (may Allah preserve him) was asked: 

We are six friends who meet every fifteen days in the house of one of us for a programme that includes Qur’an, al-Arb‘een al-Nawawiyyah (an-Nawawi’s Forty Hadeeth), Minhaaj al-Muslim, an exhortation from the one in whose house we are meeting, and Rijaal hawla ar-Rasool (Men around the Messenger – blessings and peace of Allah be upon him). We start with Qur’an and end with du‘aa’ (supplication). In between our programs, we have a piece of paper that we fill out every month, that we call a “competition schedule”; this includes a portion of Qur’an, the five daily prayers in the mosque, fasting and upholding ties of kinship. When we are consistent in filling it out, the results are good, but if we do not fill it out, the results are negative, because of neglecting to recite Qur’an. What is the Islamic ruling on this schedule? May Allah reward you with good. 

He replied: 

Praise be to Allah. What appears to me to be the case is that keeping this schedule and competing in filling it ou is a kind of innovation (bid‘ah), because it involves boasting to one another and admiring good deeds, as well as making manifest deeds which are best kept concealed, because concealing good deeds of charity, reciting Qur’an and dhikr should be furthest removed from showing off. Allah, may He be exalted, says (interpretation of the meaning): “Invoke your Lord with humility and in secret” [al-A ‘raaf 7:55] and “(This is) a mention of the mercy of your Lord to His slave Zakariya (Zachariah), When he called out his Lord (Allah) a call in secret” [Maryam 19:2-3]. And one of the seven whom Allah will shade with His shade (on the Day of Resurrection) will be “a man who gives charity and conceals it to such an extent that his left hand does not know what his right hand is giving.” See al-Bukhaari (660) and Muslim (1031). So what you should do is encourage one another to do more naafil (supererogatory) acts of obedience, and to do a great deal of that, each one doing whatever he is able to and keeping it between him and his Lord. In this manner you will be able to attain cooperation in righteousness and piety, whilst being safe from that which may nullify good deeds or detract from their reward. And Allah is the source of strength and the Guide to the straight path. And Allah knows best.

End quote from Islam Today website 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) issued a similar fatwa, as seen in Majmoo‘ al-Fataawa (16/175). 

Based on that, you everyone should be encouraged to complete the Holy Qur’an in Ramadan, without holding a contest for that, and without telling others about the number of complete readings that Allah has enabled each one to do. 

And Allah knows best.