The Bird of Hermes isn't anything, it's a metaphor Alucard uses to describe himself. Hermes, the messenger god in Greek mythology, represents Alucard himself, a servant to the Hellsing family. The second part, "eating my wings to make me tame," represents how Alucard must restrain himself due to being the Hellsings' servant.
[ In the sea without lees]
Standeth the bird of Hermes
Eating his wings variable
And maketh himself yet full stable
When all his feathers be from him gone
He standeth still here as a stone
Here is now both white and red
And all so the stone to quicken the dead
All and some without fable
Both hard and soft and malleable
Understand now well and right
And thank you God of this sight
The bird of Hermes is my name eating my wings to make me tame.