If a quote from one of your characters is more than one paragraph in length — which can happen if the character changes subject matter, or moves from an explanation to an instruction, for example (e.g., mother explains why sky is blue to child and then says, “Now go to bed.”), you need to leave the quotation marks open at the end of the first paragraph (i.e., do not insert the “close quotation” mark) and then reopen them at the start of the next paragraph. Use the “close” marks only when the speaker finishes speaking (Chicago Manual of Style Online, 13.30).
Example:
“Five little ducks went out to play,” George told his daughter. “Over the hills and far away. Mother Duck called, ‘Quack quack quack quack.’ But only four little ducks came back.
“Now, I will tell you the rest of the story after you have eaten your pate de fois gras.”
Too inelegant to tolerate
I personally find this quotation-mark construction very awkward even though it is correct. When a construction is awkward in fiction, I worry that it will lead some readers to stop and ask themselves, “What is going on here??” I will go to almost any lengths to avoid having that happen.
I would therefore rewrite the above passage thus:
“Five little ducks went out to play,” George told his daughter. “Over the hills and far away. Mother Duck called, ‘Quack quack quack quack.’ But only four little ducks came back.”
George lifted his daughter off his lap and stood her on the floor. “Now,” he said. “I will tell you the rest of the story after you have eaten your pate de fois gras.”
Note: if you are using a quote within a quote, as the writer of this story does when she has George quote Mother Duck, you use single quotes inside the double quotes. Unless you live in the U.K., in which case you do it the other way around. Or unless you are James Joyce or others of his ilk who use em dashes to signify quotations. Then you are in free-for-all land and as long as you are consistent so that the reader doesn’t get totally lost, go ahead and do whatever the hell you want: you are the creative artist.
The Militant Writer 
I have the curse of MLA training to bear.
In fiction, however, I doubt readers would put up with a character going on and on like that for too long, much less changing topics. Big block monologues are easy to identify but when the second subject/new paragraph occurs after a short quote, that is when it is most confusing for readers. Then it looks like dialogue.
Even so, where necessary, I would insert some action (e.g., He scratched his nose, looking at the sky.) and have the monologue continue as a new paragraph, subject changed or not.
Yeah, Stephen, with academic writing you can just set quotations off in blocks, which solves the problem. But that doesn’t work with dialogue in fiction!