Born on September 23, 1963, who was the first child of an astronaut?

I believe that the person you're referring to is Soviet-born Tatyana Titova, the daughter of Gherman Titov. Her father was the second human being to orbit the Earth. He did it aboard Vostok 2 in August of 1961. He still holds the record for the youngest person to reach Earth orbit, doing so at the age of 25.
For the record, Blue Origin and Virgin Galactic (commercial space carriers) have only achieved sub-orbital space.

https://x.com/NASAhistory/status/911575885714788352

OK. So, this can either be a loose answer, or a strict one. If astronaut just means someone who went to space then it is Tatiana Titova, the daughter of Soviet cosmonaut Gherman Titov.

https://en.wikipedia.org/wiki/Gherman_Titov
https://x.com/NASAhistory/status/911575885714788352

But if astronaut means someone who went by that name then it is Catherine "Katie" McDivitt born to James A. McDivitt, June 16th, 1966.

https://en.wikipedia.org/wiki/James_McDivitt
https://www.facebook.com/groups/261265912631524/posts/1338645101560261/

She holds a unique place in space exploration history as the first child born to a human who had already traveled into space (Gherman Titov became the second human in orbit during the Vostok 2 mission in August 1961). While the first person to orbit Earth, Yuri Gagarin, had his children prior to his historic 1961 flight, Tatiana's birth marked the very first time a space traveler fathered a child after returning from a mission.