What is the birthdate of Galileo Galilei's only son, Vicenzo Gamba/Gallilei?

I’m pretty sure back then they didn’t keep birth records.

There are baptismal records, but I couldn’t find anything. No google is not down as far as I know.

Perhaps there is a publication of his fathers diary in which he might have included his news.

No one knows. The baptismal record states “Vincenzo Andrea, son of Mistress Marina daughter of Andrea Gamba, father unknown, was baptized by me Father Angelo.” Compare this to the baptismal record of his eldest sister which does mention the date of birth “Virginia, daughter of Marina of Venice, born out of wedlock on August 13th, was baptized by me Giovanni Viola.”