If I got a quilt that old, I'm not sure I would be confident enough to wash it. I would get in touch with a local shop owner and find out if they can tell you of someone who does the kind of cleaning it may need. I have seen them use a sweeper and screen and just pull any dirt or dust out, to keep from getting the quilt wet.
You may want to make a few calls before putting it in water.
Getting it wet and puting it in the sun, may not be the way to get it clean. I hope that you can keep it around for another 120 years.