I assume you are referring to Flavian the Confessor, Patriarch of Constantinople from 446 to 449.
There are a couple of online hagiographies available (e.g. this one from the Orthodox Church in America). There is a much more detailed description of his life (6 pages) in Sergius of Radonezh' Lives of the Saints, available in English translation from Chrysostom Press (he is commemorated on February 18 in the Slavic churches, February 16th in the Greek).
Unfortunately, the only dates are probably ones you have already found - that he was Patriarch from 446 until 449, when he was martyred. We do know that at the time he was made Patriarch he was a presbyter at the Great Church of Constantinople (in fact, a skevophylax - responsible for the sacred property of the Church). I believe that a Church canon arising prior to that time required that one be at least 30 years old before being ordained presbyter, though I haven't been able to find it.
If, in fact, such a canon existed prior to his time, and if he met the canonical age requirement (exceptions seem to be frequent), then the best we can do would be to set an upper bound on his birth year at 416.
I read through St. Sergius account along with accounts in a shorter Greek Synaxiarion and in the even shorter Prologue of Ohrid and neither make any mention of his being old or feeble at the time he was martyred. Admittedly, this is pretty thin, but perhaps we could assume that he would be no more than 50 years old at his death in 449. This would place a lower bound of perhaps 399 AD on his birth.
So with virtually no solid information, we surmise that he was born between 399 and 416 AD.