Two points here: does your baby need a passport and, with or without one, can you fly with her within Mexico.
Yes, even as an infant, from birth, US citizens need a passport to travel outside the country and, more importantly, to return to the US. You can't be denied entry but, without identification or other proof, re-entry can be time-consuming and stressful.
To fly within Mexico, an infant who is more than 30 days old, and is not a Mexican citizen or resident, needs a passport (and where applicable, a visa). For those who are citizens/residents, it requires Clave Única de Registro de Población (CURP).
You should refer to the terms and conditions of the airline on which you'll fly; for reference, Viva Aerobus explains it all (conveniently in English, as my Spanish is solo asi asi).
Are these procedures always followed? No, but better to do it correctly, than to have difficulties and irritation, especially when travelling with a baby.
Yes, US law does requires the signature of both parents (or legal guardians). You should explain to the child's father that, rather than taking away his rights, it gives him greater protection, as it prevents the child being removed from the United States without his permission.
You might also suggest to him that he has an advantage with a notarized parental consent letter. It would show not only who he his, the father of the child, but how to reach him and under what conditions he agrees to this trip.
For you, leaving the US by air with your child would require a passport. By land, while you should have the child's passport, absent that, you need to have the birth certificate, and a good bit of patience, as you may spend more than the usual time at the border during your return.