Yes when flying with your baby you'll need proof of the babies age that is flying with you and this is certainly needed when you plan to have your baby fly in your lap.
No baby under 2 years old can fly sitting in the parents lap and any child over the age of 2 years old must fly in a separate seat on the airplane.
You can use immunization records, passports or yes the birth certificate which are all acceptable forms of identity for your baby.