The beneficial bacteria from cycling cover surfaces of your tank, rock, and equipment - there is very little of it in the water. Since the vast majority of new surface area in a tank is new rock, sand, etc. in a QT, there is little to no benefit from cycling a QT . You are likely doing heavy water changes in a QT anyway, which will control ammonia and nitrates. Biological filtration can reduce your water changes to reduce salt usage, which is beneficial in a large QT.
Here is my normal QT operation:
1) rinse QT tank with saltwater
2) perform WC on main DT (to use in QT)
3) begin drip acclimation with DT water for new stock, sometimes I will use the QT to drip acclimate the fish or coral by propping the QT so the water is all at one end until there is enough water to lay it flat again)
4) add water from DT WC to QT
4) add heater and biowheel filter with fresh clean sponge filter over intake - in this use, the sponge is only to screen big stuff from the water column and is is not intended as a biological filter. It is possible to use a sponge that was in use as a biological filter on another tank, but I do not use sponges for biological filtration.
5) move a little Matrix or live rock from sump to the biowheel box - sometimes I will skip this step if I am using a very small QT and doing water changes every 2 to 3 days.
7) if there are a lot of fish in the QT, I add Prime and an air bubbler
8) add fish, coral, etc. to QT if not added already
9) monitor tank: every 2 to 3 days max, vacuum any detritus and rinse sponge so it does not block up
10) perform a big WC weekly - 20 to 50 percent depending on amount of stock in tank and what meds are being run, if any.
Remember to replace meds in the WC water so the QT meds concentration is maintained. If the fish show any signs of distress I add Prime with the water changes to aid in protecting their slime coat. I always use DT water for the last couple of WC.
After QT operation, discard Matrix or other biological filtration, use vinegar or bleach to clean QT and all equipment then allow to fully dry before next use.