Neat quilt! Your quilting idea is a great start, but you might also want to quilt around each of he center "art" squares, and on the squares the don't have artwork, perhaps just a giant X.
Check your batting. All of them give a minimum quilting distance. This distance is in all directions.
I'm *guessing* your squares are 10" or 12". Very few battings will have a minimum quilting of 10" or less, and even then, probably not a good idea to go with the minimum ... which is why I made the above suggestion.
Another option if you are up to free motion (or have time/inclination to practice for several days first), is to just do a large meander inside all of the squares - and only up to the edge of the art squares.