It would be my thought that the mats, if not stored in a position where they would have flowing water would tend to have the bio bacteria population die back, at least some as there would be minimal flow of the ammonia and nitrite through them just laying in the pond. They wouldn't die, but would be seriously crippled. If they could be stored in a waterfall box where the water flowed upward through them, they would do much better. If they were stored in the heavy box skimmers the same benefit. For QT I use bead filters and sub pumps and can put the filter at the edge of the pond and have it working as supplemental filtration until needed for the QT.
Zone 7 A/B
Keep your words sweet. You never know when you may have to eat them.
Richard