My mamma didn't raise no quitter (didn't raise a quitter) is an expression meaning "My mother didn't raise me to be someone who gives up easily", usually used in the context of a difficult situation that has to be overcome (without quitting or running away). The speaker is using it to justify not stopping (quitting) drinking.
Both positive and negative, perhaps. He / She should perhaps quit excessive drinking, but he / she was not brought up to "quit" / to "give in" - even if the consequences were negative. (I will do anything to justify my continuing to drink, however illogical and stupid).
(Edit: Crossed with Barque).