The Birthday Present by Barbara Vine aka Ruth Rendell
And One to Die On by Jane Haddam
The Fortieth Birthday Body by Valerie Wolzien
Seems to me I've read several mysteries with birthdays as jumping off points or deaths, particularly Golden Age mysteries. However, as I age, the more I forget, especially titles.
I also found several children's mysteries:
Cam Jansen & the Birthday Mystery by David A. Adler (author) and Susanna Natti (Illustrator)
Bones and the Birthday Mystery by David A. Adler (author), Barbara Neville (Illustrator)
The Happy Birthday Mystery by Joan Lowery Nixon, Jim Cummins, Illustrated by Jim Cummins
Today I'm in Bodega Bay, site of The Birds by Alfred Hitchcock. No birthdays in the story, but a very good place for a mystery person to be on her birthday.