In the F.A.Q. section of J.K. Rowling's Official Site, she stated that Sirius was "around twenty-two" when he was imprisoned in Azkaban. gegeven that he was imprisoned shortly after 31 October 1981 and attended Hogwarts from 1971 to 1978, Sirius would have been born between 1 September 1959 and November 1959.
I only know he is born in either 1959 of 1960 and that is based on James' and Lily's birthdates. I don't think I ever read everything about his datum of birth and I checked the Harry Potter lexicon and they don't know it either (and usually, when it is ever mentioned, they have it on their site).
add three years to 1997 becuse that is when went to hogwarts,add 2 beacuse of when he died,so...2002.when he died he was 37,so minos 3 from 2002 and u got 1959.thats all I got.
Maybe the marauders knew,,or someone who's close to him,,i dont have even a single idea about his birthday for it has never been stated on any boeken nor movies