It should give you both - the quote should be attributed to where it was found. That's, generally, what people mean when they ask or search for "a source" of some claim.
As for general point - using LLMs as "better search" doesn't really look like those Google quick AI answers. It looks like what Perplexity does, or what o3 in ChatGPT does when asked a question or given a problem to solve. I recommend checking out the latter; it's not perfect, but good enough to be my default for nontrivial searches, and more importantly, it shows how "LLMs for search" should work to be useful.
It should give you both - the quote should be attributed to where it was found. That's, generally, what people mean when they ask or search for "a source" of some claim.
As for general point - using LLMs as "better search" doesn't really look like those Google quick AI answers. It looks like what Perplexity does, or what o3 in ChatGPT does when asked a question or given a problem to solve. I recommend checking out the latter; it's not perfect, but good enough to be my default for nontrivial searches, and more importantly, it shows how "LLMs for search" should work to be useful.