Quillback are a type of fish.
Like all types of fish, each quillback gives one fish fillet when caught.
Quillback can only be caught with blue and purple lures in fishing spot #12 (the shallow area with dragonflies and poles).
Quillback can be caught in small circles and therefore are harder to catch.
See Fishing List for more detailed information.
Quillback in the fishing book
The quillback is on page 22 of the fishing book:
This freshwater vacuum cleaner will eat anything. Mind your fingers!
(Note: the quillback is displayed in spot #11 in the fishing book but cannot be caught in this spot.)