First, I'd do a coral dip instead of quarantine for the corals. 5 minute dip and straight into the tank.
As for filtration, that depends on whether its a proper quarantine tank or a medicated hospital tank.
I run a true QT which is nothing more than a second display tank. I use a HOB
refugium/ skimmer combo and add some rocks and macro algae from my sump to the QT tank. I just throw the algae in there. I have a bio pad I keep in my sump for future use which I put in the QT refugium. With that I have no issue with nitrogens and a small weekly WC
is all that is needed.
After its served its purpose I clean the rocks and dry them outside for a few months before they go back into the sump. The bio pad gets cleaned and rinsed in hot hot water then washed in peroxide before it to gets dried for a couple weeks. Then back into the sump to 'cure' (build up bacteria) for its next run in the qt.
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