Yes to your proposal of washing in at least warm water--maybe hot--and drying in the dryer. You want red to do all the bleeding and shrinking it's ever going to do before you put it into a quilt. You might even want to put it through 2 or more wash cycles to be sure you have all the excess dye out of it. Use color catchers to be certain how it is behaving for you.
Of consideration, too, is how it will be handled as a completed project. Will it be gifted to someone who may not be as aware of how to care for it or will it be your own. You might always wash it in cold water but someone else might always launder in warm/hot. That could be a factor in how a quilt behaves for its owner. Prepare it for its destination.