Of all of the epithets that I’ve uncovered, I couldn’t find a single one related to birds. I was actually a bit surprised – he seems to have an epithet for everything. There are, however, several birds associated with Apollon.
Apollon’s connection to swans seems to come from several different places. His chariot is said to be pulled by several different kinds of animals, sometimes griffins, sometimes horses (though I suspect this may actually be Helios), and sometimes swans. Another story is of Cygnus, who was said to be a son of Apollo, but I suspect that his father was actually Helios. In any case, Cygnus was quite arrogant and full of himself, and he challenged his friend who loved him dearly to three nigh impossible tasks. After the tasks were completed, the friend stopped being so devoted to Cygnus, and in despair, Cygnus killed himself. His mother followed him into death. Apollo, or perhaps Helios, then put both of them into the sky as stars. A third story is about another man named Cygnus who was a wonderful musician favored by Apollon. When he died, Apollo placed him in the sky. These are two of the stories of the cygnus constellation. In addition, when Apollo was born, it is said that swans flew around the island of Delos seven times to distract Hera from Leto while she gave birth.
The raven (or perhaps crow) is another bird associated with Apollon. This story tells how ravens (or crows) were once white birds that became black, which seems to be a theme in other parts of the world. I believe there are also some Indigenous American stories that tell how crows turned from white to black. Anyway, in this story, the white raven was a messenger of Apollon. He sent this bird to check on his lover, Coronis, who was pregnant with his child (Asclepius). The bird brought back terrible news – Coronis had not been faithful. In his divine anger, Apollo turned the bird black and set Coronis on fire but at the last minute decided to save the child. The corvus constellation is a result of this anger. After her death, Apollo placed Coronis in the sky in the shape of a raven. Like many of our constellations, there are other stories. It may also be one of Apollo’s messenger ravens who thought waiting for fruit to ripen in the sun was more important than delivering water for libations.
The hawk is another bird associated with Apollon. According to Ovid, Apollon sometimes wears the plumage of a hawk. He also transformed Daedalion, the father of his lover, Chione, into a hawk after her death because he was so distraught.
I edited this post a few hours after I wrote it because more birds came to mind. As I thought of the hawk, an image of a falcon that I saw while visiting Heaven Lake in Xinjiang came to mind, so I did a quick search to see if anyone else associated the falcon with Apollon. I came across this post by Lykeia that I recommend reading.
Bearing His Light 
One of the first signs I got pointing me towards godspousery was a hawk, and I remember almost deciding it hadn’t been a thing and then reading about the Chione myth nearly a YEAR later and going “!!!!!!!!!!!!! HEY!!!!!!!!!!! THAT WAS REAL!!!!!!!”.
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