Must I perform the two votive Rak'ahs after Fajr prayer, knowing that there is a prohibition against praying after Fajr until sunrise?

The two vowed [rak'ahs] are binding upon you, except during the times when supererogatory prayers are prohibited, which are: after the Asr prayer and after the Fajr prayer. In these times, fulfilling the vow is not obligatory due to its conflict with the prohibition of the Prophet (peace be upon him), and a vow is only binding for that which is recommended (mustahabb).

Al-Dardir said in Sharh Mukhtasar Khalil: "So it is not binding for something permissible, like vowing to walk in the market, as there is no act of worship in it. And something disliked is even more so, like vowing to pray a supererogatory prayer after Asr."

The author of Rawd al-Talib said: "A vow of fasting or prayer is not valid on the day of doubt (Yawm al-Shakk) or during disliked times."

As for whether it is permissible to fulfill a disliked vow or not, there is a difference of opinion on that, and it is preferable to abandon it.