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nope... if it's for a sick fish, it doesn't need to be cycled IMO, because daily water changes should be done for any disease... if it's for new arrivals, a spare filter from another tank will suffice.
Keeping fish in your QT tank is a surefire way to add more tanks, that's how I started. Best bet is to run a secondary filter in the display tank, whether it's a HOB or a sponge filter which is the cheapest.