Here's my idea ... keep it simple!
Fussy cut some fabric with a kitty/doggy.
Sew a flap overtop, to make a 3-D kitty/doggy door.
Of course .... the rest of the quilt can be as much or as little as you want.
I might even make it a Or a kid's quilt, with a pocket of sorts inside the door ... so that the child could tuck their favourite stuffy inside.