Genter thanked the "lovely staff" at the hospital for looking after her family throughout the whole experience.
"All of you have been so generous!" she said.
The boy is Genter's and Nunns' first baby.
Genter made headlines after riding her bike to Auckland Hospital 42 weeks pregnant on Sunday morning.
She originally posted several images of herself standing beside her bike at the hospital, and another of her on the bike on Sunday.
A caption posted with the images said it was a "beautiful Sunday morning for a bike ride, to the hospital, for an induction".
Speaking at a Green Party conference in Palmerston North on Sunday, co-leader James Shaw called Genter's decision to cycle to hospital "very on brand".
"I'm very proud to be a member of a party that supports that, and proud to be living in a country where two members of the country's executive are able to have children as part of their job," Shaw said.
Genter has taken three months' leave and will return to ministerial duties on November 16.
Shaw will become acting Associate Minister of Transport and acting Associate Minister of Health while Conservation Minister Eugenie Sage will become acting Minister for Women.
The news comes a little under two months after Prime Minister Jacinda Ardern gave birth to her first child, Neve.
Ardern had Neve in Auckland Hospital on Thursday, June 21.
She has recently returned to her role as Prime Minister, following six weeks of maternity leave.