There a few places where you can get this throughout the game. Generally it will be after completing a puzzle, and Quill will stop, make noise and hold up her paw (waiting for a high five from you).
One of the first places in the game is during a puzzle in the Mire Temple level where you have to rotate a central pillar using 2 buttons on separate platforms operated by Quill and a controlled Screecher. When you press the final button on the pillar, the pillar spins around and you have to open a door on the pillar which is now facing you. When you pull the door open, Quill will be standing behind it, waiting for a high five.
To give Quill a high five, just position your blue orb (PS4 controller) over her, and press L2 / R2.