I'm pretty sure it's Sept 22, although some super-hardcore fans argue that since the Shire Reckoning Calendar is different than ours, Bilbo's "real" birthday is on Sept 13. Sort of like the argument that Jesus wasn't born on Dec 25.
It reminds me of a comicon I attended where there was a discussion of the comic Gotham By Gaslight, in which Batman fights Jack The Ripper. A fan asked if it was a time travel story, or a parallel universe, and the author replied that it was neither, but just a Batman story set in Victorian London. "So, this isn't a REAL Batman story?' asked the fan. "No," said the author, "We make ALL of them up."
The good news is that this in NO WAY interferes with my enjoyment of Bilbo's birthday. Time for some birthday cake with my daughter!