Green was born in Sydney to Australian mother Cahla and American-born father Delmas. His parents met while both were playing semi-professional basketball in Australia. The couple decided to raise a family in the north-west Sydney suburb of Castle Hill where Josh began playing basketball at the age of five when his mother started coaching him. Along with playing local basketball for the Hills Hornets as a child, Green tried an array of sports as a junior and excelled in Australian rules football, athletics, basketball, rugby, soccer and swimming. By the age of 10, he had been selected to represent his home state of New South Wales in nine separate sports. In fifth grade, Green traveled more than 12 hours by car to the outback town of Broken Hill in an attempt to gain selection for the under-12 state basketball team and was told he was good enough to be on the team but would have to wait a year or two so the older boys could play. 12 months later, he captained the under-12 NSW Metro state team to a national title. Green switched to play club basketball for Penrith in 2013 and the decision paid off when he was selection to represent the under-14 and under-16 New South Wales state teams in successive years as a bottom-ager. Green was also a prodigious Australian rules footballer in his younger years while playing locally for the Westbrook Bulldogs and would regularly attend Sydney Swans home games with his father. Green claims to have "loved" playing Australian football and has "always been a big Swans fan", naming Adam Goodes and Lance Franklin as his favourite players growing up. Such was his talent in Australian football, the Swans and fellow professional franchise Greater Western Sydney Giants offered him a place in their junior developmental academies. At the age of 13, Green accepted an offer to join the Giants' academy. However, in November 2014 the family relocated to Phoenix, Arizona for the father's work commitments and Green was subsequently forced to quit Australian football.
High school career
While in Australia, Green attended The King's School in Sydney and dominated the local basketball scene so much that he was selected to represent the New South Wales' under 16 state team at the age of 13. Later that year, the Green family relocated to Phoenix, Arizona, and in 2015, Josh enrolled at Mountain Ridge High School, where he impressed. In 2017, he transferred to the IMG Academy in Florida and became a five-star draft prospect. In the final game of his high school career, Green led IMG to a national championship victory with a 65–55 win over La Lumiere and was subsequently named MVP of the championship game.
Recruiting
Green received an offer to join the Villanova Wildcats in April 2018. In July 2018, he received an offer to join the North Carolina Tar Heels. Green was considered a top player in the 2019 recruiting class by Scout.com, Rivals.com and ESPN. On October 4, 2018, Green committed to play for the Arizona Wildcats over Kansas, North Carolina, Villanova, USC and UNLV. Green is a consensus five-star prospect out of IMG Academy in Bradenton, Florida.
College career
In October 2018, Green signed his NLI to the University of Arizona for basketball. He was selected for the 2019 McDonald's All American Game, along with fellow UA recruit Nico Mannion. In his second collegiate game, Green scored 20 points as Arizona beat Illinois 90–69. Green missed a game against UCLA on February 29 due to a lower back sprain. Green averaged 11.9 points, 4.6 rebounds, 2.6 assists and 1.5 steals per game for the Wildcats as a freshman while shooting 36% from three-point range. Following the season, he declared for the 2020 NBA draft.
Josh is the son of Cahla and Delmas Green. He has three siblings: Jay, Ky, and Maya. His father played at Oregon Tech, while both of his parents played professionally in Australia. Josh’s brother Jay currently plays at Lumberjacks.