Hopefully someone who knows/remembers more will chime in, but from what I have heard, his book mentions one hospital, and he or his sister has stated another one.
I hope Chief Engineer or someone else will correct me if this is wrong.
Jr’s sister, Maya, in a 2004 interview, stated Jr was born at Queen’s Hospital. In 2008 she said he was born at Kapiolani Hospital. Jr also mentions in his book he walked past the Kapiolani Hospital, when his father was visiting in 1971, below the room where he was born, also mentioning that the hospital had undergone renovations.