Oo Oo Oocyan
So birds do not produce blue pigments . . .
Egg of Zanzibar Bishop (Euplectes nigroventris), on the left beside egg of Zebra Finch (Taeniopygia guttata)
. . . at least not in their feathers.
While there are no known blue avian integumentary pigments, the blue-green coloration of avian eggshell is the result of biliverdin (or oocyan) pigmentation.
And in case you have an appetite for a blue omelette, read this myth-buster.