All QC fanfic would have to do to be QC fanfic would be to tell a story involving QC characters or the QC verse in a way that it's believably recognizable as such. It could be wacky, warm, serious, whatever. Doesn't have to ship or slash, although one could argue that any story is perforce about a relationship: Character with character, character with environment, Character with self. Judgment of its quality is another thing entirely. Nothing wrong with the idea of fanfic, even though it, like a lot of things, demonstrates Sturgeon's Law quite nicely.
Lot's of pro stuff out there that's basically paid-for fanfic, from Star Wars to Star Trek to Sherlock Holmes. Thousands of short stories and novels and what-have-you. The VenusPrime stuff in Arthur C. Clarke's 'verse, the expansion by other authors ofAsimov's Foundation 'Verse, the open invitation from Larry Niven to other authors to play in his Known Space 'verse. Some of it is of the "Oh, those poor trees" variety, and some of it is quite good.
By the way, I thought the fanfics linked by Carl-E and jwhouk in the fan art section were quite good.