Yep, do it to all new/recently medicated tanks. Squeezing out filter gunk from an established tank is a great way to speedup a cycle. I'd not put my main tanks filter IN the hospital tank, since it might transfer disease.
I use ceramic rings for my bio media and I use much, much more than is necessary. So, when I need to set up a hospital or quarantine tank, I just grab a few handfuls of the ceramic rings and place them into a filter of a newly set-up QT/HT.......instant cycle
Also, some people keep a sponge filter in a running tank just for this purpose. They just transfer the SF to the newly set-up tank and its good to go.