You don't have to go out and purchase special products to fix this, although you are certainly welcome to do so if you choose.
I have found the easiest way to handle bleeding fabric is to fill my bathtub with very hot water (140°), add a dab of laundry detergent, and let the fabric sit for 12 hours (usually overnight). I've never had a fabric bleed after treatment. It doesn't need multiple washings, it needs time.
Not sure how this would work if you already have white fabric sewn into the quilt, though. I usually test my fabric for bleeding before I use it in a quilt. To do this, take a piece of white scrap fabric, dampen it, and rub it gently on the questionable fabric. If any dye rubs onto the scrap, you have a bleeder.