Hi Fried Egg and welcome to TFF!
No! You'd be very disappointed if you just dropped the piece of sponge into the new tank hoping to cycle it! There's an enormous difference in effectiveness between MM (mature media) in the tank versus MM in the new filter.
You can reliably remove up to 1/3 of the biomedia (sponge or loose media) from the filter that's on the running tank and still have that tank run well. Of course, its good to go ahead and put some replacement new media in. Its not very hard to be creative on sponges with a pair of scissors.
Likewise its not that hard to turn off the new filter, use scissors on the old sponge if necessary and position the MM such that its in the water flow path just ahead of the new sponge that you want to be exposed to the bacteria. Little bits of debris containing bacteria will come off in the flow and be caught in the new sponge that comes right after.
You don't want your first stocking of fish to have gill damage from ammonia or nerve damage from nitrite, so growing the bacterial colonies with some pure household ammonia has become the method of choice. Its a very controllable way to both mature the colonies and to be able to measure and time their ability to remove ammonia and nitrite and know when the biofilter is cycled. We have a ref document and many other discussions of it here in the beginners section.
~~waterdrop~~