Should You Stop Reading the Quran during Adhan?
Question

If the Mu’adhdhin gives the call to prayer whilst I am reading Quran in the mosque, should I carry on reading and after the Adhan I can repeat the Adhan as if I am repeating after the Mu’adhdhin, or should I repeat it when the Mu’adhdhin is saying it?.
Praise be to Allah.If a person is reading Quran and the Mu’adhdhin gives the call to prayer, it is better for him to stop reading and focus on following the Mu’adhdhin, in obedience to the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “When you hear the Mu’adhdhin, say what he says” (Narrated by Muslim, 384); and delaying it will mean you miss the right time for doing it, which is the time of the Adhan .
Imam Al-Nawawi (may Allah have mercy on him) said:
When he hears the Mu’adhdhin, he should stop reading and follow the Mu’adhdhin, repeating the words of the Adhan and Iqamah ,  then go back to his reading. This is what is agreed upon by our companions. (End quote from At-Tibyan fi Adab Hamlat Al-Quran, 126)
Shaykh `Abd Al-`Aziz ibn Baz (may Allah have mercy on him) was asked: If the Mu’adhdhin gives the call to prayer whilst a person is reading Quran, is it better for him to repeat after him and say what he says, or should he focus on reading Quran on the grounds that it is better and one should give precedence to that which is better?
He replied:
The Sunnah, if one is reading Quran and hears the Adhan, is to respond to the Mu’adhdhin , following the teaching of the Prophet (blessings and peace of Allah be upon him): “When you hear the Mu’adhdhin, say what he says, then send blessings upon me, for whoever sends blessings upon me, Allah will send blessings upon him tenfold. Then ask Allah to grant me Al-Wasilah , for it is a station in Paradise which only one of the slaves of Allah will attain, and I hope that I will be the one. Whoever asks for Al-Wasilah for me, intercession will be permissible for him.” (Narrated by Muslim from the Hadith of `Abdullah ibn `Amr ibn Al-`As (may Allah be pleased with him)).
Al-Bukhari and Muslim narrated from the Hadith of Abu Sa`id Al-Khudri (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “When you hear the Mu’adhdhin, say what he says.”
Al-Bukhari narrated from Jabir ibn `Abdullah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: Whoever says when he hears the call to prayer, ‘Allahummah Rabba hadhihid-da`wat it-tammah was-salat il-qa’imah, ati Muhammadan Al-wasilata wal-fadilah, wabathhu maqaman mahmudan alladhi wa`adtahu (O Allah, Lord of this perfect Call and the Prayer to be offered, grant Muhammad the privilege and also the eminence, and resurrect him to the praised position that You have promised), will be granted my intercession on the Day of Resurrection.”
Al-Bayhaqi added with a sound chain of narration: “innaka la tukhlif Al-mi`ad (Verily You never break Your promise).” And because answering the Mu’adhdhin is a Sunnah which will be missed out if he carries on reading, whereas he will not miss out on reading because the times when he can read are plentiful. May Allah help and guide us all. (End quote from Majmu` Fatawa Ibn Baz, 10/358)
And it may be the case that what is less virtuous may become more virtuous in certain cases. For example, reading Quran is one of the best kinds of Dhikr and the Quran is the best of Dhikr. If a man is reading Quran and hears the Mu’adhdhin give the call to prayer, is it better for him to carry on reading or to answer the Mu’adhdhin?
In this case, we say: it is best to answer the Mu’adhdhin, even though the Quran is better than Dhikr. Dhikr at the appropriate time is better than reading Quran, because reading Quran is not connected to a specific time; whenever you want to you can read it, but responding to the Mu’adhdhin is connected to hearing the Adhan. (End quote from Liqa’at Al-Bab Al-Maftuh)
And Allah knows best.