Green guided North Queensland Cowboys to their first National Rugby League (NRL) premiership title in 2015.
As a player, he made 162 appearances across five different clubs during a 10-year career, before retiring in 2004.
At the time of his death, he was the head coach of Queensland in the State of Origin Series.
Queensland Police said there were "no suspicious circumstances" around Green's death and that a report will be prepared for the coroner.
Green's family said they are "devastated" by his loss.
"We cannot find the words that would come close to expressing our feeling," they added in a statement. "However we would like to extend our thanks to those who have reached out to us with their love and support.
"Paul was loved by so many and we know that this news will generate immense interest. However, at this time we ask for privacy."
Australia Rugby League commission chief Peter V'landys paid a glowing tribute to Green and praised his "brilliant" abilities as a coach and player.
"Paul became one of the few to make a very successful transition into coaching," said V'landys.
"I had the pleasure of sitting on the NRL Competition Committee and found Paul to be a passionate, smart and witty individual. Our condolences go to his family and to his many friends."
Green, who was born in Brisbane, broke a number of records during his six-year spell as Cowboys coach, including having led the side to their best ever defensive campaign in 2016.
He left the Cowboys in 2020 following a club-record 87 wins in 167 games.
Cowboys chairman Lewis Ramsay credits Green with having reinvented the club.
"Paul's arrival as head coach transformed our club from finals contenders to an immediate premiership force, culminating in the historic 2015 Grand Final victory," said Ramsay.
"We will forever remember Paul as one of the greatest contributors in Cowboys history."
A number of Australian rugby league clubs have paid tribute to Green on social media.