[3] 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.

[5] Green switched to play club basketball for Penrith in 2013 and the decision paid off when he was selected to represent the under-14 and under-16 New South Wales state teams in successive years as a bottom-ager.

[6] Green was also a prodigious Australian rules footballer in his younger years[7][8][9] 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 rules football[10][11][12] and has "always been a big Swans fan", naming Adam Goodes and Lance Franklin as his favourite players growing up.

[17] While in Sydney, Green attended The King's School in Parramatta 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.

[18][19] Later that year, the Green family relocated to Phoenix, Arizona, and in 2015, Josh enrolled at Mountain Ridge High School, where he impressed.

On 4 October 2018, Green committed to play for the Arizona Wildcats over Kansas, North Carolina, Villanova, USC and UNLV.

[34] On 27 December, after a stint in health and safety protocols, Green had a career-high 10 assists in a 132–117 win against the Portland Trail Blazers.

[37] On 21 April 2022, during the Mavericks' first round playoff series against the Utah Jazz, Green recorded 12 points, three rebounds, six assists and two steals in a 126–118 game 3 win.

[46] Green was named in the 24-man squad selected to represent the Australian national team in FIBA World Cup Qualifiers against Qatar and Kazakhstan in September 2018.

[3] On 6 February 2019, Green was named in the 23-man squad selected to represent Australia's Under-19 national team at the 2019 FIBA Under-19 Basketball World Cup.

He has three siblings, two brothers, Jay, who played for the Northern Arizona Lumberjacks,[51] Ky and a younger sister named Maya.