I agree that the QT needs to cycle, but the cycle will be faster because the amount of ammonia you introduce will be much smaller and controlled. If you throw a piece of raw shrimp into a completely clean tank with only a sponge filter, it won't take long for the bacteria to build up to reduce the ammonia. Once that happens you can add a fish.
The reason DT's take so long to cycle is that we introduce a very large volume of dead or dying stuff, with very little 'good bacteria' to convert it. The bacteria levels take much longer to build up to those levels, whereas in a small & sterile QT there's very little ammonia to begin with so you don't need as many bacteria and this makes it faster.
You definitely want to monitor the cycle in a QT tank and make sure to either feed it or put in a raw shrimp to introduce some ammonia, but I'd expect to see the tank ready to go within 2 weeks. From then on make sure to keep some filter media or a couple of pieces of LR in your sump, so you can start up the QT anytime you want without worrying about waiting for it to cycle.