"It's a very exciting time in anybody's life and it certainly is in mine," said Williamson.
Williamson missed the second test against the West Indies to be with Raheem as the pair awaited the birth, with the Black Caps batsman having flown back and forth between Wellington and Tauranga before returning to Tauranga to be with his wife ahead of the test.
Black Caps coach Gary Stead said at the time that the whole team was supportive of Williamson.
"Family comes first. Kane's not the first player to miss a test match because of an impending birth and he won't be the last," Stead said.
"Worst case scenario Kane misses some matches.
"As a dad, as a parent, you only get that opportunity once in your life to be there for your (first) child's birth and I know that it's important for Kane too.
"At the end of the day we play cricket and other things are much more important."
Raheem is a qualified nurse, who studied at the University of Bristol.
The pair have largely kept out of the limelight away from the cricket arena, but have been a couple since at least late 2015.
Williamson was not named in the Black Caps' squad to play Pakistan in the first Twenty20 tomorrow night at Eden Park, but was tenatively named in the squad for the clashes on Sunday and Tuesday.
The Black Caps' next test match is on Boxing Day.
