This question can be handled two ways. Either you have a quilt already constructed like an antique quilt that you want to keep safe, or how to construct a quilt that will not call apart.
When making a quilt, using strong threads on the binding, quilting, and any key areas is the best guard against issues later. Also using the right fabric. High quality fabrics found in quilt shops are always better than discount stores. I understand we like to cut corners on cost from time to time, but fabric is not the place to cut it. Never buy from discount stores. Always stick to your trusted local quilt fabric shops and well known name fabric lines.
When washing a quilt you suspect may fall apart, put it in a machine that it will fit in comfortably. If this is not your machine, go find an industrial machine at a laundry mat without a center agitator. Put it on the lightest setting and avoid hot water. Use a light detergent, and put it through a second rinse. When drying use a low setting or air dry it on a line. A small amount of fabric softener should be fine.