Do not know whether the quilt can be washed with water. The clothes on which we sleep, and wear must be washed in water. If something cannot be washed with water, it is better to avoid it.
Since you have said that you have back pain, use a mattress specifically for the duration of menstruation. And use a separate set during the non-menstruating period. The bed used while menstruating can be kept aside after sprinkling with water.