Yes, I know it’s Gene Roddenberry in our world, but in the TOS episode The Man Trap, Sulu says this to Yeoman Rand:
“May the great bird of the galaxy bless your planet!”
You can see it in this clip: https://www.youtube.com/watch?v=lk3tM4Ns3LY
There is also this Okudagram that appeared in two TNG episodes (The Naked Now and Conspiracy) and was clear enough on-screen to make it less than just a joke:
So it appears that there is something known as The Great Bird of the Galaxy, it has a human head, possibly a Starfleet delta, and it’s used in a saying that may or may not be an actual blessing.
So what was it? Was it an Academy joke? Was it a real religion? Obviously it was more than just something Sulu made up since there’s an entry for it on the Enterprise D’s computer (which seems like an odd thing to have if it’s an in-joke).
I agree, there are a lot of inside jokes. This, however, was, as I said, clear enough on screen to count as more than that.
Here it is from Conspiracy:
There’s an entry in Memory Alpha with a few other references, one being Peter David novels where it is an actual creature.
Unfortunately, they don’t give any more information than I have already given, aside from the Peter David novels, which aren’t considered part of Star Trek canon.