The date is corroborated by the Historia Augusta:
Ipse Antoninus Pius natus est XIII. kal. Oct. Flavio Domitiano XII. et Cornelio Dolabella consulibus in villa Lanuvina.
A bit on his accession from the OCD (3rd ed.):
His links with the Annii Veri, combined with his wealth, poularity, and character, led Hadrian to choose him as adoptive son and successor on the death of L. Aelius Caesar. Given imperium and the tribunicia potestas on 25 February 138, he became Imperator T. Aelius Aurelius Antoninus Caesar and at Hadrian’s wish adopted both the young son of L. Aelius (the future L. Verus) and his nephew by marriage M. Annius Verus (Marcus Aurelius). His accession at Hadrian’s death, 10 July 138, was warmly welcomed by the senate, which overcame its reluctance to deify Hadrian at Antoninus’ insistence and named him Pius in acknowledgment of his loyalty. His wife Faustina was named Augusta and his only surviving child, also Annia Galeria Fuastina (2), was betrothed to M. Aurelius Caesar, his nephew and elder adoptive son.