Following his 2007 first grade debut, he played two seasons for the St George Illawarra Dragons in a variety of backline positions, his preferred position being left centre. He joined the Bulldogs for the 2009 NRL season in the hope of securing a permanent starting spot in the centres. Injury permitting, Josh achieved this throughout the season. He made his representative debut for Country in the 2007 City vs Country Origin match. In 2008, he played wing in the Australian Prime Minister's XIII side in its match against Papua New Guinea.
Canterbury-Bankstown
Morris moved to the Canterbury-Bankstown Bulldogs for the 2009 NRL season. In the 2009 State of Origin series Josh was named in the 40-man New South Wales preliminary squad for State of Origin. He was then selected for game 2 and 3, having missed the first game through injury. Morris made his State of Origin debut for NSW 31 years to the day that his dad Steve had made his only test appearance for Australia in 1978. Morris finished as Canterbury's top try scorer in 2009 as the club finished the regular season in second place. Morris played in the club's preliminary final defeat against arch rivals Parramatta at ANZ Stadium. He was selected in the 24-man squad for the Gilette Four Nations Tour at the end of the 2009 season and made his international debut for Australia against France at Charlety Stadium in Paris on 7 November, scoring two tries playing inside his brother Brett, who also scored twice. In round 3 of the 2010 NRL season, he scored the first 4-try haul of his first grade career in Canterbury's 60–14 drubbing of the Sydney Roosters. Morris was selected to play on the right wing against Wales in the 2011 Rugby League Four Nations. At the 2012 Dally M Awards Morris was named the NRL's centre of the year. Morris played in the 2012 NRL grand final loss to Melbourne at Centre and nearly scored a try which would have brought Canterbury back into the game but the ball was knocked out of his hands by Melbourne fullback Billy Slater. Morris played for Canterbury in the 2014 NRL Grand Final loss to Souths at centre. On 10 July 2018, Morris signed a two-year deal to join Cronulla starting in 2019 due to the salary cap crisis at Canterbury where the club needed to offload players. Morris was in talks with the Gold Coast Titans but ultimately decided to join Cronulla as he would have a chance to play under coach Shane Flanagan.
Cronulla-Sutherland
Morris made his first appearance for Cronulla in round 1 of the 2019 NRL season against Newcastle which ended in a 14–8 loss. Morris scored his first and second try for the club against North Queensland in round 3 as Cronulla won 42–16. Morris was selected to play for New South Wales in the 2019 State of Origin series. Morris played in Game 1 which New South Wales lost 18–14 at Suncorp Stadium. Morris was subsequently dropped from the New South Wales team by coach Brad Fittler for the second game. In round 16 against Brisbane, Morris scored a hat-trick but it was ultimately not enough as Cronulla lost the match 24–22 at Shark Park. In Round 23 against the New Zealand Warriors, Morris scored 2 tries as Cronulla won the match 42–16. Morris played 23 games for Cronulla in the 2019 NRL season as the club finished 7th on the table and qualified for the finals. Morris scored a try in the club's elimination final defeat against Manly at Brookvale Oval. Morris finished the year as Cronulla's top try scorer with 15 tries. In February 2020, it was reported by the media that Morris had handed in a transfer request to the Cronulla board citing that he wished to join his brother Brett at the Sydney Roosters. Morris spoke to the media saying “I spoke to the boys at Cronulla and I know the timing isn’t ideal. It’s not because I’m unhappy with any of the boys at the Sharks, But this is a unique opportunity. My brother isn’t just any old mate, this is the bloke who I started this journey with, “Now another opportunity has presented itself and I’d regret it for the rest of my life if I didn’t try and make it happen". On 11 March 2020, Morris request to be released by Cronulla-Sutherland to join the Sydney Roosters was granted. The Cronulla club released a statement saying "The 33-year-old will be released following the Sharks round two match against the Melbourne Storm. "After much internal discussion and following lengthy negotiations with the Sydney Roosters, the release request has been granted ultimately due to circumstances surrounding the Sharks salary cap, a situation which has been impacted by previous infringements". A transfer of up to $75,000 was also included by the Sydney Roosters to help alleviate the Sharks’ well-publicised salary cap issues.
Sydney Roosters
In round 3 of the 2020 NRL season, Morris made his debut for the Sydney Roosters in their 28-12 win over rivals South Sydney.
