Yes, you can quilt with invisible thread in the top, I don't recommend it in the bobbin. I'm using it now to piece, not quilt ... but I'm piecing a landscape quilt so sometimes my piecing is over multiple layers so technically it's kind of quilting. I've had no problems with breakage but I did have to play with my tension and used a size 70 needle.
Use Polyester monofilament - not Nylon. Can't speak from experience but folks here and elsewhere have said Nylon can: discolor with age, become brittle with age, and melt under an iron or hot dryer. Polyester isn't bothered by any of these. Both Sulky and YLI make good Poly Monofilaments in both clear and "smoke".
Or you can use white to quilt on the colored blocks.