Quick, who had six shutouts last season, made 28 saves to notch his 16th career shutout.
Dustin Brown and Kyle Clifford tallied for the Kings, who won their third straight.
Mike Smith stopped 20-of-22 shots for Phoenix, which lost its second consecutive game.
Brown scored 56 seconds into the second period on a power play to put the Kings on the board. Anze Kopitarcontrolled the puck in the right circle and dished it behind the net to Mike Richards, who found Brown in the crease to jam it home.
The Kings capitalized on a 3-on-2 fastbreak when Kevin Westgarth made a cross ice pass to Clifford, who wristed a shot from the left circle past Smith to give Los Angeles a 2-0 lead 15:23 into the second period.
Phoenix had a power play at the end of the game, but Quick came up with some big stops to preserve the shutout.
The Kings return home to play the Stars on Saturday while the Coyotes hit the road to play the Ducks on Sunday...Quick's shutout was his 100th career win.