Cows have bacteria in their rumen which produce methane. As a curiosity, I ddg’d “do pigeons fart”, and according to pigeonpedia they do not because they have short digestive tracts lacking gas producing bacteria.

https://pigeonpedia.com/do-pigeons-fart/

Bird poop tends to be dispersed (with some exceptions), and so will tend to decompose aerobically, which does not produce methane.