First and foremost, it is not necessary that the Holy Qur’an 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 Qur’an do prove the existence and also mention about the government of Hazrat al-Hujjat (a.t.f.s.). Eminent scholars have listed such verses in their books. Among them Allama al-Majlisi (r.a.) deserves special mention for having brought numerous such verses in his book “Behar al-Anwar”, some of which are mentioned hereunder:
وَلَقَدْ كَتَبْنَا فِی الزَّبُورِ مِنْ بَعْدِ الذِّكْرِ أَنَّ الْأَرْضَ یَرِثُهَا عِبَادِیَ الصَّالِحُونَ
“And certainly We wrote in the Book after the reminder that (as for) the land, My righteous servants shall inherit it.”(Surah al-Anbiya’, 21:105)
هُوَ الَّذِی أَرْسَلَ رَسُولَهُ بِالْهُدَىٰ وَدِینِ الْحَقِّ لِیُظْهِرَهُ عَلَى الدِّینِ كُلِّهِ وَلَوْ كَرِهَ الْمُشْرِكُونَ
“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.” (Surah at-Tawbah, 9: 33)
وَعَدَ اللَّهُ الَّذِینَ آمَنُوا مِنْكُمْ وَعَمِلُوا الصَّالِحَاتِ لَیَسْتَخْلِفَنَّهُمْ فِی الْأَرْضِ كَمَا اسْتَخْلَفَ الَّذِینَ مِنْ قَبْلِهِمْ وَلَیُمَكِّنَنَّ لَهُمْ دِینَهُمُ الَّذِی ارْتَضَىٰ لَهُمْ وَلَیُبَدِّلَنَّهُمْ مِنْ بَعْدِ خَوْفِهِمْ أَمْنًا ۚ یَعْبُدُونَنِی لَا یُشْرِكُونَ بِی شَیْئًا ۚ وَمَنْ كَفَرَ بَعْدَ ذَٰلِكَ فَأُولَٰئِكَ هُمُ الْفَاسِقُونَ
“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.” (Surah an-Nur, 24:55)
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.).
وَنُرِیدُ أَنْ نَمُنَّ عَلَى الَّذِینَ اسْتُضْعِفُوا فِی الْأَرْضِ وَنَجْعَلَهُمْ أَئِمَّةً وَنَجْعَلَهُمُ الْوَارِثِینَ
“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.” (Surah al-Qasas 28: 5)
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.”
لَنْ تُغْنِیَ عَنْهُمْ أَمْوَالُهُمْ وَلَا أَوْلَادُهُمْ مِنَ اللَّهِ شَیْئًا ۚ أُولَٰئِكَ أَصْحَابُ النَّارِ ۖ هُمْ فِیهَا خَالِدُونَ
“Know that Allah gives life to the earth after its death; indeed, We have made the communications clear to you that you may understand.” (Surah al-Hadid, 58: 17)
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 Qa’im (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 Qur’an about the occultation and reappearance of Imam (a.s.). The discerning should therefore refer to this reliable book.
Source: 14 Questions about Imam az-Zaman Volume 1