What does (the expression) "to stir the paint" mean?
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.
This is a metaphor for a man and a woman having vaginal sex while the woman is menstruating. Because- and only because- Rimmer has specifically requested a more detailed explanation, I'll continue. Consider this fair warning that the following explanation is sexually explicit.
One stirs a can of paint with a stick (in order to ensure that the pigment is distributed evenly throughout, tho that's not relevant to this answer). In this sexual metaphor, the woman's (red) menstrual blood is represented by the "paint," and the man's penis is represented by the "stick." Thus to "stir the paint" is to stick one's penis (the stick) into a menstruating woman's vagina, allowing it to be covered with menstrual blood (the paint).
EDIT: This phrase is highly non-standard; it's a joke rather than a common idiom. If you're looking for usage history or source, I think Christopher's link to the Urban Dictionary is as good as you're going to get.
To stir the paint is to have sex with a woman while she's on her period.
Source: http://www.urbandictionary.com/define.php?term=stir+the+paint