Can You Pray Sunnah after Fajr?
Question

Is there any Sunnah prayer after Fajr prayer?
Praise be to Allah.Fajr Sunnah prayer
There is no Sunnah prayer after Fajr prayer . Before Fajr there is a Sunnah ratibah prayer (established Sunnah) which is two rak’ahs. This is the most firmly established of the Sunnah ratibah prayers. 
The Prophet (peace and blessings of Allah be upon him) never failed to offer this prayer, whether he was at home or travelling. 
`Aishah (may Allah be pleased with her) said: “The Prophet (peace and blessings of Allah be upon him) did not adhere more firmly to any nafl prayer than the two rak’ahs of Fajr.” (Narrated by al-Bukhari, 1163; Muslim, 724) 
Virtues of Fajr Sunnah prayer
Concerning the virtue of the two rak`ahs before Fajr prayer, the Prophet (peace and blessings of Allah be upon him) said: “The two rak'ahs of Fajr are better than this world and everything in it.” (Narrated by Muslim, 725)
What to recite in the Sunnah prayer before Fajr
It is Sunnah to recite in these two rak'ahs al-Kafirun and al-Ikhlas , because Muslim (726) narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) recited in the two rak’ahs of Fajr: “Qul ya ayyuhal-kafirun” and “Qul huwa Allahu Ahad.” 
Can we make up the Sunnah prayer before Fajr?
The one who misses the Sunnah prayer before Fajr may offer it after Fajr prayer . 
The evidence for that is the report narrated by al-Tirmidhi (422) and Abu Dawud (1267) from Qays ibn ‘Amr who said: The Messenger of Allah (peace and blessings of Allah be upon him) came out when the iqamah for prayer had been given, and I prayed Fajr with him, then the Prophet (peace and blessings of Allah be upon him) finished and found me praying. He said, “Take it easy, O Qays. Two prayers together?” I said, “O Messenger of Allah, I did not pray the two rak’ahs before Fajr.” He said, “No, then.” According to the version narrated by Abu Dawud, the Messenger of Allah (peace and blessings of Allah be upon him) remained silent. This hadith was classed as sahih by al-Albani in Sahih al-Tirmidhi. 
Al-Khattabi said: 
“This shows that the one who misses the two rak'ahs before the obligatory prayer may offer them after the prayer, before the sun rises.” (`Awn al-Ma’bud)
It says in Tuhfat al-Ahwadhi: 
“Two prayers together?” is a question in the sense of denouncing, i.e., two obligatory prayers at the time of one obligatory prayer? Because there is no nafl prayer after Fajr. 
“No, then.” 
It should be noted that the words of the Prophet (peace and blessings of Allah be upon him), “No, then,” mean: there is nothing wrong with you offering these two rak'ahs at that time. This is indicated by the report of Abu Dawud, where it says, “and the Messenger of Allah (peace and blessings of Allah be upon him) remained silent.” And ‘Ata ibn Abi Rabah narrated that a man of the Ansar said: “He did not say anything to him.” Al-'Iraqi said: its isnad is hasan. And a version narrated by Ibn Abi Shaybah says: “He did not tell him to do that or tell him not to do it.” And a version narrated by Ibn Hibban says: “he did not denounce him.” The reports explain one another.” 
And Allah knows best.