Were any popes or patriarchs converts to Christianity? The Biblical Peter was an adult when Jesus began his ministry, and thus ranks among the earliest converts to Christianity.  Roman Catholics hold that Peter was also the first pope.  Are any subsequent popes known or suspected to have been converts?

More generally, among the historical popes and patriarchs of the apostolic sees, were any of them (other than the founding apostles themselves) believed to be converts?
According to Catholic tradition, rarely questioned, the Roman church was led by the apostle Peter, who appointed his successor as bishop of Rome, and that Rome had an unbroken series of bishops down to the present day. At first, the bishops of Rome were not known as 'popes', but even the earliest Roman bishops could now be referred to as popes. According to this  tradition, the earliest popes, at least during the first century,  must have been converts to Christianity. 

Francis A. Sullivan SJ says, in From Apostles to Bishops: The Development of the Episcopacy in the Early Church, page 100, there is general agreement among scholars that the structure of ministry in the church of Rome at this time would have had a group of presbyters sharing leadership, perhaps with a differentiation of roles among them, but with no one bishop in charge. He says (ibid page 15) the broad consensus among scholars, including most Catholic ones, is that only during the course of the second century did the threefold structure become generally the rule, with a bishop, assisted by presbyters, presiding over the church. The import of this is that the Church was well established before the first bishop of Rome was appointed or elected. We can no longer assume that history necessitated early popes being Christian converts as adults. 

Nevertheless, it is known that at least some popes were pagan converts to Christianity.  Pope Symmachus (498-514) is known to have been born a pagan in Sardinia and baptised after his arrival in Rome.

Patriarchs became part of the Church hierarchy in the fourth century, with the the Council of Nicaea in 325 CE identifying the bishops of Rome, Alexandria, Antioch and Jerusalem as Metropolitans with regional powers (provision was made later for Constantinople and Carthage). I doubt that any comprehensive list of all Patriarchs and of their former beliefs, if any, now exists.