Quick played travel youth hockey with the Mid Fairfield youth hockey association out of The Darien Ice Rink. He carried his team to win the National Championships several years in a row. His team also featured forwards Sean Backman, John Mori, and Augie DiMarzo, who all continue to play professional hockey.
Quick played at Hamden High School in Hamden, Connecticut before transferring to Avon Old Farms where he was named to the 2002 New Haven Register All-Area Ice Hockey Team. He led Avon Old Farms to two straight New England Prep Championships in his junior and senior seasons. Quick had nine shutouts during his senior year.
He played collegiately for the University of Massachusetts Amherst. As a freshman, he earned his first victory by a score of 4–2 on October 15, 2005, in his first start against Clarkson University. He also appeared in one playoff game as a freshman, which the Minutemen lost 4–1 to Boston University. Quick scored his first goal January 6, 2007 against Merrimack College. In the 4-2 win, it was a delayed penalty, empty net, unassisted game winning goal at 9:31 of the second period. While playing for the University of Massachusetts Amherst, he led the Minutemen to their first NCAA Men’s Ice Hockey Championship appearance. In his first ever NCAA Tournament game Quick shutout Clarkson University, stopping all 33 shots he faced, in a 1–0 overtime victory. He appeared in 5 playoff games, and posted a .944 save percentage. Quick still holds many records in UMass-Amherst hockey history.
After losing in his first professional start, with the ECHL‘s Reading Royals, Quick made a statistically unlikely goal during a shutout in his second start, scoring an empty-net goal against the Pensacola Ice Pilots on October 27, 2007, at 19:25 in the third period.
Los Angeles Kings
Quick began the 2008–09 season with the Kings’ American Hockey League (AHL) affiliate, the Manchester Monarchs, sharing time with Jonathan Bernier. He was called up to the Kings on December 16, after goaltender Erik Ersbergsuffered a groin injury. Quick played his first NHL game on December 6, 2007 against the Buffalo Sabres in an 8–2 win. Quick earned his first career NHL shutout on December 23, 2008, defeating the Columbus Blue Jackets inColumbus, Ohio. He also received the NHL’s Third Star of the Week for the week ending December 28. In three games, Quick posted a 2–1–0 record, with both victories coming via shutout, a 0.67 GAA, and a .958 save %. On February 9, 2009, Quick was given the first star of the week by the NHL by stopping 95 of 100 shots in a 3 game road win streak which launched the Kings back into the 2008–09 playoff picture. He finished the season with a 21–18–2 record, with a 2.48 goals against average and 91.4% (.914) save percentage. He earned his first career NHL Playoff shutout on April 17, 2011 against San Jose Sharks stopping all 34 shots for a 4–0 win.
On January 1, 2010 Quick was named to the United States Mens Hockey Team for the 2010 Winter Olympics in Vancouver, British Columbia as the team’s third goaltender. He received a Silver medal after USA lost to Canada 3–2 in the Gold medal game.
On April 25, 2012 Quick was nominated as a Vezina Trophy finalist, along with Henrik Lundqvist (winner) and Pekka Rinne. Quick was also named a 2011-12 NHL Second Team All-Star. Quick led the league with 10 shutouts (also a Kings franchise record), had the second-lowest goals-against average with 1.95, and posted a 35–21–13 record. Quick gained much attention during the 2012 Stanley Cup Playoffs as the Kings defeated the No. 1 seed Vancouver Canucks in five games, followed by the Kings franchise’s first-ever four-game playoff series sweep against the No. 2 seed St. Louis Blues.
On May 30, 2012 Quick set a new NHL playoff record, winning his 11th consecutive road game dating back to the 2011 post-season, surpassing the old mark of 10 by Billy Smith of the New York Islanders, and continued adding to the record. The streak ran from April 16, 2011 at San Jose to June 2, 2012 at New Jersey (12 games). The team qualified for the conference final for the second time in its 45-year history and first time since 1993. Los Angeles ended its 19-year drought and made it to the Stanley Cup Finals after defeating the number three seeded Phoenix Coyotes in Game 5 of the Western Conference final.  Quick was awarded the Conn Smythe Trophy following the Los Angeles Kings first Stanley Cup victory over the New Jersey Devils, 6-1. Los Angeles took the series 4-2. He posted a 16-4 record with a 1.41 GAA, a .946 save %, and 3 shutouts.
On June 28, 2012 the Los Angeles Kings and Quick agreed, in principal, on a ten-year contract. This will keep him under contract until the 2023 season.