Attalid dynasty
Gaius Julius Quadratus Bassus was born in Pergamon to a family related to the Attalid dynasty and the Galatian tetrarchs. His father was Gaius Julius Bassus, who was Proconsul of Bithynia in 100 to 101. He is known to have had at least one son, Gaius Julius Bassus, who was suffect consul in 139.