I recently heard in an online lecture by Dr. Martin Blinder that the Romans had considered Apollo's birthday to be December 25th. I'm having trouble finding a source that validates this statement
One can refer to 25 December as Apollo's 'birthday' but he certainly didn't have a monopoly on the day.
Apollo, being associated with (among many other things) the sun / light, was one of many pagan gods who was linked to December 25th. This is connected to the winter solstice and the rebirth of the sun (hence the date being after December 21st - at one point December 23rd was celebrated but this was later changed to December 25th). For the Romans, though, Apollo was less important than he was for the Greeks. The Romans already had a deity for the sun (although Apollo became more popular at some point) and thus there was an
antagonism between Apollo and Sol. In its form of Sol Indiges, the latter was apparently an age-old Roman deity, whose functional sphere came to overlap with that of Greek Apollo as an identification of the ‘sun’, when the latter identification became prominent in Rome in the Augustan period
Source: Michael Lipka, 'Roman Gods
See also 'Sol Invictus'