Aconcio was born around 1520 in Trento, Italy, or possibly the nearby town of Ossana. He was the son of Gerolamo Aconcio and his wife, Oliana. He studied law and later became a notary in 1548 when he was admitted to the Collegio dei Notai of Trento. In 1549 he entered the service of Count Francesco Landriano, a courtier serving the emperor, Charles V in Vienna. Aconcio remained with Landriano for seven years and then around 1556 became secretary to Cardinal Madruzzo, the imperial governor in Milan.