The phrase stir the paint is not standard slang. It cannot be found in this UK dictionary of slang, this slang site, this other slang dictionary, or this published dictionary of slang. However, there are similar phrases which have sexual meanings. According to the Cassell’s Dictionary of Slang, these are some of the slang uses of stir as a verb:

stir fudge–1960s, to perform anal intercourse

stir it up–1950s, to masturbate

stir one’s stew, stir the batter, stir the sauce–1950s+, see stir it up

stir shit–go out of one’s way to make trouble

stir the porridge–1980s, Australia, to have sexual intercourse with a woman immediately after she has had intercourse with another man, esp. used of the final man in a gang-rape

stir the possum–to create a disturbance

stir the stew–1900-1910s, to have sexual intercourse

So there is a highly sexual connotation to stir that has been around since the 1900s. It seems likely that stir the paint is similarly sexual, although it is not documented before the Family Guy episode aired. Thus it is also likely that Family Guy coined the phrase, but did so based on these preexisting meanings. I would hazard a guess that the “paint” is menstrual blood.

It is similarly possible that stir the paint is not sexual at all, but instead refers to some dynamic between the two parents in which Lois will not let Peter do tiny things like stir paint, but given the nature of the humor in Family Guy I doubt this.