Reference: Fataawa al-Lajnah ad-Daa.imah lil-Buhooth al-‘Ilmiyyah wal-Iftaa. – Volume 7, Page 244, Question No. 3, Fatwa No.7845
Question: Do the two [Sunnah] rak’ahs of Salaat al-Fajr suffice in place of the [two rak’ahs of] tahiyyatul-masjid? And if a person was to perform both the [two rak’ahs of] tahiyyatul-masjid and the two [Sunnah] rak’ahs of Salaat al-Fajr, has he sinned?
Response: The two [Sunnah] rak’ahs of Salaat al-Fajr suffice for itself as well as the [two rak’ahs of] tahiyyatul-masjid; and this is better and preferred. And if he performed the [two rak’ahs of] tahiyyatul-masjid first and then the two [Sunnah] rak’ahs of Salaat al-Fajr, then there is no harm [in having done this].
And with Allaah lies all success and may Allaah send prayers and salutations upon our Prophet Muhammad ﷺ and his family and his companions.