It is actually neither. Kellan Lutz's, the actor that played Emmett, his birthday is March 15th, 1985. The only information that was given on Emmett's birthday is that he was born in 1915.
Emmett Cullen's birthday is on March 15, 1915, not 1895 or 1935.
I think it was 1935 but I'm not sure
I don't think it's mentioned, but in New Moon, when Edward and Bella are going to the Cullens' house for her birthday party, Edward says something along the lines of "the last birthday any of us had was Emmett in 1935"
Carlisle was born in the 1640s Jasper was born in 1843 Esme was born in 1895 Edward and Alice were born in 1901 Rosalie and Emmett were born in 1915
The books don't say when Esme's birthday is but she was born in 1895 in Columbus, Ohio
Matthew Ridgway was born on March 3, 1895.
Jesse Baker was born on March 4, 1895.
Milt Gross was born on March 4, 1895.
Shemp Howard was born on March 4, 1895.
Betty Schade was born on March 27, 1895.
Fredric Wertham was born on March 20, 1895.
Zlatko Baloković was born on March 21, 1895.
Encarnacion Alzona was born on March 23, 1895.
Christian Herter was born on March 28, 1895.