I kid you not -- there are over a dozen ways to sign birthday.
Your best bet is to interact with plenty of native Deaf adults in
your local region and see which version they are signing.
Until you discover which version of birthday is most popular in your
area (there might be several versions used in the same region) a
fairly safe way is to just combine the signs "BIRTH" and "DAY."
If and when you get "corrected" just smile and thank them for
letting you know.
But hey, since you asked me, I'll show you how I sign "birthday."
I touch the tip of my middle finger to my chin, then I touch it to
my heart.
BIRTHDAY:



NOTES:
People who are left hand dominant ("lefties") can do a mirror image.
They don't have to end "over the heart." They can end on the other
side of the chest.
As a student or newbie you should NOT argue with your local Deaf
regarding how to do this sign. And certainly don't argue with your
local ASL instructor (unless you want a bad grade eh?). Just
be flexible and "When in Rome do as the Romans do."
The generic version of "birthday" is to just combine one of the
versions of "BIRTH" with one of the versions of "DAY."
BIRTHDAY: BIRTH + DAY:




Also see: BIRTH
Also see: DAY
You can learn
American Sign Language (ASL) online at American Sign Language University ™
ASL resources by Lifeprint.com © Dr. William Vicars