Is it valid to pray two rak'ahs for Fajr immediately after the athan, then engage in remembrance (dhikr) and recitation of the Quran, followed by praying the two rak'ahs of the Subh prayer?

The time for Fajr begins with the appearance of the true dawn (al-Fajr al-Sadiq), and hearing the adhan is not a prerequisite. However, if the adhan is called by a righteous muezzin who is knowledgeable of the prayer times, then it is permissible to rely on it. In this case, there is no harm in performing the two rak'ahs of the Fajr sunnah prayer and then delaying the obligatory Fajr prayer after engaging in some remembrances (adhkar) and reciting the Qur'an.