What does the underside of the quilt look like? If you could post a photo of the wrong side, that would help.
It appears to me that this is a pieced log cabin quilt, but that one of the fabrics used in the piecing was a fabric that was meant to be used as a "cheater". If this is the case, you would ignore those printed black stitches in the cheater fabric.
Are you thinking of quilting this by hand or by machine?