The girl, named in reports as Terrie Kirby, died along with friends named as Alexandra Binns, 18, Ryan Beal, 20, and Brandon Brown, 20.
The quad bike was in a crash on the A6201 between Upton and Hemsworth in West Yorkshire early on Sunday.
Two men who were in the car were questioned about the crash and bailed.
Three of the people died at the scene while one of the men died in hospital from his injuries.
Friends and family members went on to social media to express their condolences and post messages of sympathy, and one said Terrie had died on her birthday.
Ashleigh Lynch wrote on Facebook: "I don't actually know what to do with myself.  Absolutely wounded, my heart's actually melting.
"Why does it have to the best ones? My best friend out of it all. And on her 16th birthday.
"You're my angel. My beautiful Alexandra, Beal...  and Drea will show you how to have a good time."
Jeorgia Lynch added: "Can't believe it, doesn't seem true. Going to miss you Ryan Beal, Terrie Lynch. Fly high, don't party too hard up there, beautiful angels.
"Love you Alexandra Binns... too young, taken too soon."
Ray Henshaw, principal of Minsthorpe Community College where Terrie Kirkby was a pupil and Alexander Binns and Ryan Beal had previously been taught, said the deaths were "devastating".
He said: "We are trying to help our young people come to terms with it, the understandable grief and loss.
"The sense of devastation is [because of] the age of these young people, the tragic loss and the fact they had so much of their lives left ahead of them."
In a statement, glazing firm Safestyle, where Ryan Beal worked as a fitter, said staff were in mourning.
The firm said: "We are deeply saddened and shocked by the news.
"Our heartfelt condolences go out to the family and friends of Ryan."
West Yorkshire Police said the quad bike was unregistered and believe the Nissan was travelling towards Hemsworth on the A6201 when it collided with the bike.
At a press conference earlier, Det Ch Insp Richard Homes said the car was seen parked on the bypass prior to the incident.
The A6201 is still closed with rush-hour traffic being diverted away from the scene.
Four people who died when a sports car collided with the quad bike they were on had been celebrating the 16th birthday of the youngest victim.