I would NOT transfer a filter from a tank used as a hospital tank into a main display tank because you could also be taking bad bacteria or introducing Ich into the other tank. These bad bacteria and parasites can live within filter media and so you might as well not quarenteen any of your fish. Keep in mind that your tank does not automatically have Ich in it, it must be introduced into a tank from another source. This is one of the principles behind using a quarenteen tank in the first place; to notice and identify any potential problems before adding a fish into a main tank.
You can keep a hospital tank up and running easily by cycling it first (for this, you can borrow your main tank's filter in order to 'clone' the tank as well as using some substrate from the other tank in the new tank. Bacteria will be in the substrate as well). For times that your hospital/Q-tank will be empty for extended periods, I suggest simply 'feeding' the tank (adding some flake food, maybe every other day) in order to feed the bacteria or by keeping some ghost shrimp in it. These shrimp are very cheap (usually used as feeders) and do not live very long so they are not going to be a big committment for you as far as keeping them healthy and what not. They will eat dang near anything so you can just feed them like you would if there were fish in there. They also add very little messy wastes and so you could most likely get away with changing the water on this tank every other week rather than every week. You can also use the water that you take out of the main tank when doing a water change to add into the hospital/Q-tank. This will suppliment some nitrites and a few nitrates to the tank. Keep in mind though that the good bacteria does not live in the open water in any quantity big enough to cycle or keep a tank cycled.
I HIGHLY suggest not transferring filter media from one tank to another though. I understand the principle behind it and it does make sense, but like i said, when you put the borrowed filter back to its origonal tank, you could be adding more than you think. I have only heard of this method being used when wanting to cycle a new tank or even in preparation of a planned tank purchase in the future (some people will buy the filtration system first and hook it up to an existing tank in order to 'seed' it)