Question: Why is Brian trying to convince Peter to move on from Quagmire's pocket-dialed conversation when Brian and Quagmire hate other?
Answer: It is not that Brian and Quagmire both hate each other; it is actually just Quagmire who hates Brian, but Brian does not hate Quagmire. Brian tells Peter to move on from Quagmire's pocket dial because confronting him about it may result in him ruining his friendship and cause heartache for everyone involved.
Answer: The other answer is wrong, Brian has shown a dislike for Quagmire - specifically because of Quagmire's dislike of him. Most likely Brian, who has shown feelings for Lois throughout the show too, is acting on his own feelings of guilt.