Marcus,

the only ancient source I’ve ever heard of for Tiberius’ Olympic victory is the inscription on a statue of Olympian Zeus in honour of the event. I haven’t read the inscription, but it is apparently to be found in V. Ehrenberg & A.H.M. Jones, Documents Illustrating the Reigns of Augustus and Tiberius, Oxford, 1955, no. 78. The year was 1 A.D., it was during Tiberius’ voluntary exile on Rhodes. As you know, in ancient times the winner of the quadriga race wasn’t the person who actually raced the thing, but the owner of the team. Tiberius was tough, but no Commodius, so it is very likely he didn’t drive the quadriga himself.

The 2 sources for it that I have are: chp. II of the biography of Tiberius by Ernst Kornemann (Tiberius, W. Kohlhammer Verlag, Stuttgart, 1960) and an article by Jean-Paul Thuillier, “Tibère et Néron, champions olympiques” in Les Collections de l'Histoire magazine, July/September 2008 issue.