When looking for a sound that you can imagine, but have a hard time finding a good source for, sometimes it helps to think of concrete objects that might make a similar sound like the one you're looking for. Like the old frying-bacon-to-make-rain-trick
Another idea – specifically for creature sound design – is to approach it from a pseudo-zoological or even anatomical angle. What kind of sounds would your specific griffin utter in the wild? Does its head look like a giant chicken, or more lizard-like? Questions like that will provide you with clues. When you almost get the feeling that you
found the sound, it was there all along and you just grabbed it, that's when you know you got something good. Then with time you'll find more ways to do foley stuff if you want to record, or get a good ear for library sounds and how to edit everything together to create something that's yours.
I can't remember what exact combination of sounds we used for the griffins in our audio adaption of Cornelia Funke's
Dragon Rider: A Griffin's Feather (you can find it on Audible, read by Marc Thompson), but I think we used some of the excellent Boom Library creature sounds, tuned them around a bit and mixed them with other animal sounds (not just birds). There's a couple different griffins in the book and we tried giving each a slightly different sound. For the wings I recall recording different cloth flaps to get the feathers right...