I am joining the Fajr prayer in 2nd rak'at, qunut is being performed, I join the imam and follow accordingly. After tasleem has been said, I continue for my missed rak'at. Must I repeat qunut again in my 2nd rak'at ?
Simply said the 2nd Rak'a for the Imam and the congregation is your first raka'a, as you joined them during the qunut and before moving to ruku'.
So once they have completed the congregation by tasleem or salam you go ahead to complete your prayer by performing your second raka'a. Now whether you will perform the qunut or not is a matter of your madhhab, some consider it sunnah so it would be highly recommended to do it, while others say it is mustahab or don't approve it at all for the sobh prayer (fajr fard). In the first case you should repeat the qunut in the last you are (either) free to decide to do it or (you should) leave it.
AFAIK only the shafi'i madhhab consider it sunnah for fajr fard, the Maliki considers it mustahab, the hanafi only approve qunut for witr prayer.