I see no harm in a UV on a qt tank or just about any other tank, with the exception of one that is being treated with certain UV sensitive medications.
I, personally, would not waste an expensive unit on a qt tank. In my qt tanks, I generally will salt and/or treat as needed, but the tank is often not even cycled. I change 100% of the water daily - leaving behind any water bourne nasties in the last batch of water - hense there is little need for a UV. A UV on a main tank is wonderful - and it will help protect your new fish as well as your old fish as you mix them together - just in case there is anything that may attempt to transfer through the water from one to another.
For the UV lamp is best used for removing the vectors between fish. This means that for one fish to infect another fish the parasite or bacteria or whatever must pass from the infected fish, through the water to the second fish. The UV unit, properly running, will reduce or eliminate this direct vector from one fish to another.
The UV will not remove anything that is actively on or in the fish - it can only work on that which is water bourne - that which passes through the unit and is exposed to the UV rays. This is why it will not cure a fish that is already carrying a parasitic load, but it can help reduce reinfection once the fish is cleared of problems. It does a HUGE amount in reducing the transfer of problems from fish to fish - like a personal force field.
Salt, because it comes in direct contact with the fish in the tank, is invaluable during quarantine. Unlike the UV, salt CAN remove parasites that are actively on and infecting a fish. Salt dips can strip away many slime dwelling parasites. Salt bathes (as in 0.3% for 2 weeks) can kill off a good deal of the nasties. I would not replace the salt treatment with a UV.