Yellowed lines can be spot cleaned with a mild solution of OxiClean and water. Rinse well, by hand and dry out on the lawn. You can also make a saltwater solution to clean the whole quilt. Dilute 1/4 cup sea salt per one gallon of warm water. Rinse and lawn dry. Lemon juice in warm water can work as well. Rinse and lawn dry.
Do not use bleach. Do not dry clean. Overexposure to bright sunlight may bleach out bright colors.
The reason to spread the quilt out on the lawn is so that the fabric is not pulled or stressed by hanging on a line.