So he was born on Feb 11 the year before the British empire adopted the Gregorian calendar. After the change, his birthday was Feb 22. Feb 22 is the date that we generally remember. That makes sense because George himself wouldn't have any personal memory of the Julian (often called "old style" at the time) date.

However, as you'd expect of such a situation, a lot of people at the time knew the old style Feb 11 date, so he was known to acknowledge both dates to people that came up to him. Somewhat famously, the last year of his life he publicly celebrated his birthday on Feb 11 while away from home, and then upon returning to Mount Vernon he celebrated it again on Feb 22.