Does Fred Flintstone give birth to three babies? If yes how?

This happens in season 2 episode 3, "The Missing Bus", and no, Fred does not give birth to babies. The fandom page for the episode notes the ending as:

When Fred successfully assists one of the mothers of his charges in a search for her missing boy, she goes into labor and requires immediate transport on Fred's bus to hospital, where she gives birth to triplets, and Fred is a hero.

So, it is the mother who Fred has on the bus who goes into labour and who has the triplets. Fred is just there because he drove her. He also happens to be their Godparent. Presumably Fred has a hand in naming them because of that and him being the hero that drove them there.

Nurse: Oh, Mr. Flintstone would you like to see your godchildren now?

Wilma: You mean they were a pair?

Fred: Better than a pair, three of a kind.

Wilma: Oh, they're cute. What are their names?

Fred: What else? Fred, Fred, and Fred.

Wilma: Three Freds? But isn't that confusing?

Fred: Yes, considering that they're all girls.

_{The Flintstones, Season 2 Episode 3, "The Missing Bus"}