It looks to be the seam lines that have frayed out? If you used a consistent 1/4 inch seam throughout, then that fabric was poor quality . I would contact where you bought it. So disappointing because it is a darling quilt!
I might cut out some appliqués with a fusible on the back and placed them over the frayed spots. See if they look okay before ironing them down. If they do I might edge them with a thread that matches your backing fabric and stitch around them.