Do the verses of the Holy Quran prove the existence and reappearance of Hazrat Mahdi (a.t.f.s.)?
First and foremost, it is not necessary that the Holy Quran explains in detail all the concepts. Given that it actually deals with many things, so it does not get involved with the finer aspects and intricacies.
Yet, there is an affirmative reply to this question as well. Many verses of the Majestic Quran do prove the existence and also mention about the government of Hazrat-e-Hujjat (a.t.f.s.). Eminent scholars have listed such verses in their books. Among them Allama Majlisi (r.a.) deserves special mention for having brought numerous such verses in his book “Behaar al-Anwaar”, some of which are mentioned hereunder:
وَ لَقَدْ كَتَبْنَا فِى الزَّبُوْرِ مِنْ بَعْدِ الذِّكْرِ اَنَّ الْاَرْضَ يَرِثُهَا عِبَادِىَ الصَّالِحُوْنَ
“And certainly We wrote in the Book after the reminder that (as for) the land, My righteous servants shall inherit it.”
هُوَ الَّذِىْ اَرْسَلَ رَسُوْلَهُ بِالْهُدٰى وَدِيْنِ الْحَقِّ لِيُظْهِرَهُ عَلَى الدِّيْنِ كُلِّهِ وَلَوْ كَرِهَ الْمُشْرِكُوْنَ
“He it is Who sent His Apostle with guidance and the religion of truth that He might cause it to prevail over all religions, though the polytheists may be averse.”
وَعَدَ اللهُ الَّذِيْنَ اٰمَنُوْا مِنْكُمْ وَعَمِلُوْا الصَّالِحَاتِ لَيَسْتَخْلِفَنَّهُمْ فِىْ الْاَرْضِ كَمَا اسْتَخْلَفَ الَّذِيْنَ مِنْ قَبْلِهِمْ، وَ لَيُمَكِّنَنَّ لَهُمْ دِيْنَهُمُ الَّذِىْ ارْتَضٰى لَهُمْ وَلَيُبَدِّلَنَّهُمْ مِنْم بَعْدِ خَوْفِهِمْ اَمْنًا، يَعْبُدُوْنَنِىْ لاَ يُشْرِكُوْنَ بِىْ شَيْئًا، وَ مَنْ كَفَرَ بَعْدَ ذٰلِكَ فَاُوْلٰئِكَ هُمُ الْفٰسِقُوْنَ
“Allah has promised to those of you who believe and do good that He will most certainly make them rulers in the earth as He made rulers those before them, and that He will most certainly establish for them their religion which He has chosen for them, and that He will most certainly, after their fear, give them security in exchange; they shall serve Me, not associating aught with Me.”
Elucidating this verse in his book “al-Ghaibah”, Shaikh Tusi (r.a.) testifies to it being revealed about the occultation, reappearance and the supporters of Imam (a.t.f.s.).
وَ نُرِيْدُ اَنْ نَّمُنَّ عَلَى الَّذِيْنَ اسْتُضْعِفُوْا فِى الْاَرْضِ وَ نَجْعَلَهُمْ اَئِمَّةً وَّ نَجْعَلَهُمُ الْوٰرِثِيْنَ
“And We desire to bestow a favour upon those who were deemed weak in the earth and to make them the Imams and to make them the heirs.”
In the same book “al-Ghaibah”, Shaikh Tusi (r.a.) reports a tradition from Ameerul Momineen Ali (a.s.), expounding this verse as follows:
“When the earth becomes restricted and a killing field for the Holy Prophet’s (s.a.w.a.) progeny, then Allah will send Mahdi, to degrade the enemies and elevate the friends.”
اِعْلَمُوْآ اَنَّ اللهَ يُحْىِ الْاَرْضَ بَعْدَ مَوْتِهَا، قَدْ بَيَّنَّا لَكُمُ الْاٰيٰتِ لَعَلَّكُمْ تَعْقِلُوْنَ
“Know that Allah gives life to the earth after its death; indeed, We have made the communications clear to you that you may understand.”
Explaining this verse Sheikh Tusi (r.a.) narrates in his book “al-Ghaibah” from Ibn Abbas as follows: “The giving of life to the earth after its death in this verse refers to the Qaim (a.t.f.s.) from the progeny of Muhammad (s.a.w.a.). After reappearance, he will purify the earth from all evil and enliven it through justice and equity.”
Likewise Allama Majlisi (r.a.) brings 60 verses from the Holy Quran about the occultation and reappearance of Imam (a.s.). The discerning should therefore refer to this reliable book.
 Surah Anbiya (21), Verse 105
 Surah Tauba (9), Verse 33
 Surah Nur (24), Verse 55
 Surah Qasas (28), Verse 5
 Surah Hadid (58), Verse 17