Transferring water from an established tank to a "brand new" tank will do virtually nothing in terms of speeding up the cycling process... nor will squeezing your sponges from an established tank into a non-cycled tank. Vacuuming and transfering the slug from an established tank may help somewhat... but again, will do almost nothing very useful (plus if you keep a clean and light/moderately stocked tank, its difficult to get a substantial amount of sludge to begin with).
I would suggest leaving extra sponges within your current filtration in your established tanks. When and if the time comes, remove the sponge(s) from your established tank and transfer it to your Q-tank.
My 66 gallon Malawi tank has 2 AC300 and a fluval 404 running on it. My AC300 have multiple sponges in them (just sponges I took from my various collection of AC200, minis, etc.), and my fluval 404 is packed to the brim with bio media (rings, etc.). If I need to get my 15 gallon Q-tank running, I just pop on an AC-mini remove a sponge from one of my AC300s and transfer a few rings from my fluval 404, and the Q-tank is ready to go.
On another note, since my 66 gallon tank is over-filtered, removing a couple of sponges and some bio-media does nothing to the bio-filtering capacity of my tank. There is never any detectable indication that my ammonia or nitrite levels have raised. Another reason to overfilter your tanks!
HTH
-Richer