Is it permissible for me to pray the two rak'ahs of Fajr during the Adhan? And thank you.

If the muezzin gives the call to prayer (adhan) after the time for Fajr begins, and the true dawn (al-Fajr al-Sadiq) has become apparent, then praying during the adhan is valid, whether it is an obligatory (fard) prayer or a supererogatory (nafila) one. However, it is preferable to wait until the adhan finishes, repeat after the muezzin, then recite the prescribed remembrances (adhkar), then pray the pre-Fajr sunnah prayer, and finally, the obligatory prayer.