I've shipped tons of yarn and fabric and cloth diapers... The very best method I have found is to get a large flat rate box and put a garbage bag in it open at the top. Fold the quilt to the correct dimension and set it inside. It will be taller than the box, but that's ok. Use a vacuum to suck all the air out of the garbage bag, twist the top and tape it shut. You will be amazed at how small it gets. Everything I have shipped this way fluffs back up at the other side no problem.
and definitely get insurance and delivery confirmation!!!