According to Fantastic Beasts and Where to Find Them, the griffin has the head, front legs, and wings of an eagle. So it would probably sound like an eagle.
Note that the sounds eagles make aren't very dramatic, so most eagles in film (or any bird of prey, in general) has the screech of a Red-Tailed Hawk dubbed over.
Eagles (Bald Eagle and Golden Eagle)
But hey, since this is a magic griffin, I guess you can make it sound like whatever you want. So you can have a long, piercing hawk-like screech instead of the short yapping an eagle makes, I suppose.
Tim the Enchanter