Can I pray (Sunnah)of the Fajr after the Fajr?
All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad  is His slave and Messenger.
  is His slave and Messenger.
If you find a congregation performing the obligatory Fajr prayer and you have not yet performed the Sunnah of Fajr, then you have to join them and later perform the Sunnah of Fajr. The evidence for this is the Hadeeth of Qays ibn Fahd who said: “The Prophet  saw me performing two Rak'ahs after finishing the Fajr prayer. He asked: “What are these two Rak'ahs?' "What is this prayer?" I said: “O Messenger of Allah! I did not perform the two Rak'ahs that precede the prayer of Fajr, so I performed them now.” [Ahmad, Abu Dawood and at-Tirmithi]
  saw me performing two Rak'ahs after finishing the Fajr prayer. He asked: “What are these two Rak'ahs?' "What is this prayer?" I said: “O Messenger of Allah! I did not perform the two Rak'ahs that precede the prayer of Fajr, so I performed them now.” [Ahmad, Abu Dawood and at-Tirmithi]
The Prophet  did not make any comment. So, the silence of the Prophet
  did not make any comment. So, the silence of the Prophet  proves that it is permitted. However, it is likeable to perform these two Sunnah Rak'ahs if one misses performing them before Fajr after the sunrise. The Prophet
  proves that it is permitted. However, it is likeable to perform these two Sunnah Rak'ahs if one misses performing them before Fajr after the sunrise. The Prophet  said: “Whoever missed performing the two Rak'ahs of Fajr, he should perform them after sunrise.” [At-Tirmithi]
  said: “Whoever missed performing the two Rak'ahs of Fajr, he should perform them after sunrise.” [At-Tirmithi]
Allah knows best.
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