Quizzlebee's does not actually exist in the Harry Potter novels or movies, but came into existence due to its overly English-sounding name and a certain, je ne sais quoi, that makes the reader instantly think of magic, Wizards, and disapparating.
Still don't believe in Quizzlebee's? Try to imagine Michael Caine saying it...
Hagrid: What are you going to get, Harry?
Harry Potter: Either a roast beef with unicornraddish, or a dragon, lettuce, and tomato with cool, ranch dressing.
Hagrid: Aye, betwixt you and me I'd rather be subjected to the Cruciatus curse than eat at Quizzlebee's. I'll be outside, methinks.
Harry Potter: Here's 3 Galleons, go grab some greasy rubbish at Death Eaters (another fictional Harry Potter eatery).