TheFreeDictionary could be clearer; essentially, what it means is something more like the Wiktionary definition:
give a damn (third-person singular simple present gives a damn, present participle giving a damn, simple past gave a damn, past participle given a damn)
To be concerned about, have an interest in, to care (about something).
Frankly, my dear, I don't give a damn.
I'd question the assertion that it's always used in the negative; the Wiktionary entry lists a non-negative construction, and so others surely exist, such as your example.
To TRomano's point about the confusion: I believe the phonologically reduced n't in couldn't causes the expression's meaning to be carried in the intonation rather than the lexicogrammar.
That is, because the n't is so difficult to hear sometimes, the following sentences can have the same meaning, depending on the tone:
I couldn't care less
I could care less
Having said that, I'm not sure the change is complete for give a damn, or even that it's necessarily the same thing. I'd want to see more examples of them.
For instance, COCA provides the following results:
(A) -not/n't GIVE a damn 252
(B) not/n't GIVE a damn 547
(C) not/n't * GIVE a damn 24
Note that in the above results,
* means one word and not any number of characters as it common RE.1 Additionally, GIVE means the lemma - all forms of - give (given, giving, gives)
Firstly, what this shows - even if there could be distant, non-negative constructions - is that overwhelmingly it is used in the negative, but not always.
Consider the excerpt from (A) (above, apologies for the small text here's a direct link), in particular the ones without negation; lines 3 and 15. These examples don't have any distant or proximal negation, and their meanings are clear.
For (3), it is we are smart enough to care. For (15), it is How am I going to care about selling someone a T-1 line?
Of course, language changes, and spoken language so much more rapidly than written language. If give a damn adopts the same meaning whether it is negated or not, it is still in the process of doing so.
Also note that having a negative connotation is not the same as being negated; things can have a negative connotation ("have bad axiological relations") and not be negated. And, looking at the corpus data for hoot and shit (links below), the same is true for them; they retain both a negated and non-negated meaning.
She does not give a damn about her job = She gives a damn about her job
tl;dr: No, at the very least, the above clauses are not equivalent yet, and I'm not sure they're in the process of becoming equivalent. This applies to hoot and shit as well.
Here are links for each of the searches: