If you can't find the mattress cover at a store that sells moving and packing
supplies, then you might be able to find a tarpaulin ("tarp") at a hardware
store. A tarpaulin is a very large piece of plastic or water-proof canvas. You
can wrap it around the mattress and fasten the ends together with duct tape or
some other tape. Here in the U.S., I've seen plastic tarps in orange or blue or
see-through clear plastic.