You could, indeed do that. Look at "quilt as you go" instructions to get an idea of how it's possible. I would really have to want it to tackle it, though. I don't think it would be easy. It might look odd if you don't do it well and find a complimentary fabric for the addition.
Whether it's a solid or a print, it will need to blend with the colors and patterns of the existing quilt.
bkay