I think you mean you are cycling a Quarantine tank not a Hospital tank. A hospital tank is for treating sick fish and usually should not be a cycled tank. There are two reasons for this.
1. Some medications that you may have to use will kill off the bacteria.
2. What may be left behind. If you treat fish in an H tank and are not 100% certain of what the problem is and the fish dies, you don't know the cause. As a result you must completely sterilize everything in the H tank, including filter media, as it may contain the cause and using it will just create more sick fish. When you sterilize the H tank and its contents, you will kill the bacteria. Even if you know the cause, you may still need to sterilize.
A good strong bleach solution is what I use for sterilizing.
