Anne Shirley was born in March 1865.

In Chapter 13 of Anne of Avonlea, Diana mentions that Anne's birthday is in March.

"I was just on my way over to invite you to help me celebrate my birthday on Saturday," said Anne.

"Your birthday? But your birthday was in March!"

Anne's birth year of 1865 is estimated at the Chronology of the Anne Books, based on the approximate ages of Anne's children, Anne's marriage to Gilbert at age 25 (Anne's House of Dreams, Ch 2), and clues in the Montgomery works. 

In Budge Wilson's 2008 prequel Before Green Gables, it's mentioned that Anne is expected to be born in mid-March (Chapter 5) and that she was born on a Friday (Chapter 6). Anne states her birthdate as March 5th in Chapter 34.

"That's a long time ago. Six years ago on the fifth of March I was born. That's my birthday."

There is some dispute whether this date should exist in Anne's continuity because L.M. Montgomery did not write the work.  However, Beyond Green Gables was a prequel specifically authorized by the LMM Heirs as an "official" prequel to Anne of Green Gables.  Whether the March 5th date should be considered canon is up to the perspective of the reader.