I would try rubbing a solution of Synthrapol and water (maybe 1:10) into the marks with a soft toothbrush, then wash the entire quilt in the washing machine with Synthrapol. Synthrapol will lift unset dye particles and keep them suspended in water to be rinsed away. Many dyes these days are chemicals that are set by means of other chemicals, not by heat, so I would think there is a good chance you could remove the line with Synthrapol. You might need to repeat the process once or twice.
Synthrapol is widely available online (check Amazon, for example) and sold in quilt stores that cater to fabric dyers.
Edit: Chalk dyes are notorious for being hard to get out of fabric. White chalk is much easier.