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)