Her birthday is unknown since she was in an insane asylum when she was changed and does not remember anything about her human life. Her full name is Mary Alice Brandon Cullen (New Moon page 400).
From Alice: Full name: Mary Alice Brandon Cullen tarehe of Birth: Some tarehe in 1901(not absolutely sure) From Ashley: Full name: Ashley Greene tarehe of birth: 21 February 1987, Jacksonville, Florida, USA
Noone knows when her birthday is. It states in New Moon that she couldn't find the file au whatever that has her Birthday on it. All we know is that she was born in (1901). Her real name is Mary-Alice-Brandon