I have never bothered trying to have a hospital tank cycled. I find it is usually a waste. Whether or not I succeed in having a fish recover or not, I always bleach the tank and all its contents when I am done with it- this includes any filters and media.
The reason I work this way is threefold. First, I do not want anything nasty left over, especially if I have not succeeded in saving the patient. Second, a fair number of medication will kill off the beneficial bacteria thereby rendering any attempt to have a cycled tank useless. Third, I rely on regular waterchanges- normally 25% every other day, but more if needed, to keep the ammonia levels down and the water quality up.
Other considerations in a hospital tank are:
-Have it be bare bottom so one can see what the fish is eating as well as what it is pooping.
-Use sponge type filtration to minimize current so the fish doesn't have to expend energy on this. Throwing out a sponge and using a new one the next time is inexpensive.
-Add a few fake plants to provide a bit of cover with the intent to reduce stress levels on the fish (fake can be bleached real ones I would want to throw out).
Q tanks are a bit different. While they certainly can be cycled, one must be prepared to medicate them which may result in trashing the bacteria etc. My normal practice is to have a couple of smaller tanks cycled and occupied with some fish I can easily move out when I need to press the tank into Q service. Failing that I can usually swipe enough media/decor from several other tanks and some plants from my excess plant bin that I can use to help with bio-filtration. I always assume everything in a Q tank may end up having to undergo the same bleach sterilization as an H tank.
One never knows in advance how long fish will end up being in quarentine nor what problems may arise. What I do know is I want to see fish free of any signs of illness for at least 30 consecutive days and for recent imports even longer to be sure any dangers have passed.
This is what has worked best for me over the years, however it is certainly not the only way to do things.
