Hawks breath through the nares or nostrils at the part of the beak up near the face.
You can see the nasal holes in this picture
and a close up here
They also pant to help cool themselves and that is done through the mouth.
Hawks get some moisture from the prey animals they eat but they also can land to drink from water sources. They pick the water up in their beaks and swallow it.
You can see one drinking here
Hope this helps you!
Let me see if I can find another for you