The Quahog Veterinary Clinic is where Brian is taken after he is hit by a car in "Life of Brian". On the way, Brian is forced to share an ambulance with a chicken that refuses to talk to Peter, noting he does not have a good reputation in the chicken community.
The vet informs the Griffin Family that there isn't anything he can do and allows them to say their goodbyes as Brian recovers long enough to tell them that he also loves them before he dies.
Brian goes to the clinic after becoming ill in "Once Bitten", where he is told his problems stem from his drinking. To treat him, Peter is given suppositories to administer to Brian.
When a depressed Brian eats chocolate at Valentine's Day in "Boy (Dog) Meets Girl (Dog)", he accidentally poisons himself and ends up at the clinic. There, he meets another dog named Ellie and they hit it off.