"give a damn" is defined as follows in the Cambridge Idioms Dictionary, 2nd edition:
give a damn (about somebody/something) (slang)
to be interested or involved He sent his son to parochial school because he believes that those schools give a damn.
Usage notes: also used with verbs like could and might to mean someone is not concerned about something: A significant portion of kids in class could give a damn.
Some of the other comments call the use of the phrase "could give a damn" to mean "does not give a damn" incorrect and have made analogy to "I couldn't care less" ("damn" might also be replaced by "rat's ass", "shit", or "fuck", incidentally). Logically, I don't think this phrase is truly analogous. It works because "could" (in the present) points to counterfactual situations -- consider the difference between "I can speak Spanish" and "I could speak Spanish if I would have paid more attention in Mr. Garcia's class." † Or consider the difference between answering an invitation "Yes, I can go skiing on Saturday" and "I could go skiing on Saturday, but that place is too expensive" -- you can't really switch "can" and "could" in these examples.
With that said, I think it would be helpful to pull up some actual examples to prove that this phrase is far and away most commonly used to imply the negative (i.e., to not care). I just searched Google News; the examples seem to skew toward colorful and opinionated rants. That's not too surprising considering the phrase, so I hope you won't read too much into the choice of stories to take these from.
“I did not feel any better. I didn’t feel any passion from him,” Dash, a frequent Fox News contributor, went on. “I felt like he could give a shit [...]"
Talking Points Memo, "Fox News Host: Obama ‘Could Give A Shit’ About The Threat Of Terrorism"
"I have never voted," Knight told the audience. "I'm not here to represent the Republican party. Quite frankly, I could give a damn about the Republicans. And on the other hand I don't give a damn about the Democrats, either."
Real Clear Politics, "Bobby Knight at Trump Rally: I Could Give a Damn About Republicans or Democrats, 'At Heart He's A Great American'"
The person going to this game is the type that gets up at seven in the morning to watch Arsenal vs. Chelsea at Fado in Center City Philadelphia. He probably has a Theo Walcott jersey that he also wears to the noon pickup game at YSC Sports. This guy is a diehard soccer fan who could give a rat's ass about the Philadelphia Union and probably thinks that "the MLS sucks."
That's the guy. He comes to the Tuesday night futsal game wearing a Barcelona tracksuit, claims to be a striker, then whiffs on his first shot attempt from five yards out.
Philly Voice, "Miami's 'El Clasico' a blatant cash grab and everything that's wrong with American soccer"
For 10 years, I have carried that document in my briefcase, waiting for just one liberal in the mainstream media to ask me about it. That has never happened. It’s like the liberals consigned that document to the howling winds of Siberia. They could give a damn about that document, just as they could give a damn when Obama in September 2009 sold out Poland to Putin. They only care now, when the Russians might have sold out their Hillary.
The Spectator, "When Liberal Doves Become Russia Hawks"
Since the Edinburgh-bred singer is coming off her first stretch of tour dates in a while, her voice is still a bit raw, and she keeps cracking herself up into coughing fits. She is 49, but “could give a flying fuck” about age and appearances, and tends to go for men who value intelligence over beauty. “But it’s easy for me to say that,” she notes. “I wasn’t born with a typical cliche of a female body—didn’t have the big tits, didn’t have the huge arse, wasn’t curvy. So I’ve never really relied on my body.” She shrugs and flicks a piece of salmon into her mouth with her chopsticks. “Who cares?”
Vice, "Shirley Manson Asked Me Out for Sushi for Some Reason"
By contrast, I could not find any examples of the phrase meaning "to care."
† "had paid" is the more standard way to say this but it seems stilted to me in this example.