Quagmire and Brian should be friends. Quagmire has no reason to hate Brian. It seems like everytime Brian and Quagmire meet, Brian greets him and Quagmire brushes him off.
Brian innocently tries to make conversation with him and Quagmire always gives him the cold shoulder. Then Brian would see something Quagmire is doing and asks him a question about whatever it is, and Brian even misinterpets it and Quagmire yells at him.
Then Brian would tell a joke or make a statement, and Quagmire takes it too personally. This Quagmire hates Brian plot never should've happened Quagmire is really the problem, not Brian.