With 10 gallons, you have a lot of options. You can always just change water out of your display and into the 10 gallon tank when you want new fish, along with some filter media. If your QT was larger, this is not always easy. You can keep it running too. Sponge filters are cheap and easy to seed and dispose of and you can use filter bags for HOB filters or even synthetic quit batting as media.
Coral QT is harder. Some corals suffer without fish, but you can kinda do the same thing, or just keep it running. If you are QTing the corals for fish parasites, then you have a long term job coming. If just for coral parasites, then the timeframe can be lower. Have some ammonia/ammonium on hand to feed the corals.
If you are worried about fish parasites in the tank, then you have to QT inverts too, so don't forget about them.
I more isolate and prepare fish for captive live (eating well and not skittish) rather than QT, so take all that I said for what it is worth. I usually advise people to setup their display with real live rock teeming with life so that if any diseases get through, the tomonts don't hardly stand a chance in the substrate with all of the critters looking for a meal. However, you can do both.