I'm sure the reader has seen a bird move its head about, no? When a bird is curious or wary, it moves its head about quickly, almost jerkily.
I have thought of using these words: bob, wrench, flick, tilt, wag, shift, waggle, pump, snap, flip, and dip. I like a few of them, but I was hoping some of you fine people could provide a word with better specificity.
In a sentence:
The crow flicked its head about, wary of my presence, gawking at me with beady eyes.