Quate jumped out of his car to look over the bridge.
"There was a guy in the water yelling for help, I ran across the bridge to the other side and saw a second person, a woman, float to the surface, face up and unconscious."
The off-duty cop scrambled down the slope and stepped into "freezing water", quickly finding himself out of his depth.
"The current carried me about 10 metres. There was a branch hanging down and I managed to hook my right arm over it. I reached out and grabbed the woman and pulled her over."
The man, who Quate described as fatigued and distressed, managed to pull himself over and grab the branch.
"We were clinging there. I was trying to calm him down and keep my head above water."
Some men at the water's edge were able to throw a strop to Quate, who secured it to the unconscious woman.
All three were hauled to the shallows.
Quate performed CPR on the woman and succeeded in clearing water from the woman's lungs.
An off-duty nurse supervised resuscitation attempts until paramedics arrived.
It was the second time he had performed CPR in his 18-year career - both times he was off-duty.
The woman was taken to hospital but died there on Monday. The man, who was driving their car, was also taken to hospital with non life-threatening injuries.
Quate said he was "very sorry to hear" the woman did not survive, and his thoughts were with the family.
"I'm very grateful to the guys who pulled us out and took over CPR. We couldn't have got out of there without them."
The car was floated by members of the Police National Dive Squad and recovered on Thursday, described by squad OC senior sergeant Bruce Adams as "a difficult operation".
Quate's boss, Eastern District commander superintendent Tania Kura, said while the crash had a tragic outcome, Quate was to be congratulated.
"He acted with great courage and presence of mind in very difficult circumstances," she said. "I extend my sympathies to the family concerned but no one could have done more than Scott did to try to bring about a happier outcome.
"His selfless actions reflect the determination of our Police staff to make this a safer country for all."