it works just like dry rock cycling, its variable based on conditions and they break down similar to this:
sponge placed in clean water, zero pre cycling, not anything added except for natural contaminants both as the bac seed source the ammonia break down, could be 4 months of submersion before decent nitrifiers are present to digest small testable amounts of ammonia. they do self seed and self attain feed, says the millennia
sponge placed in matured reef, allowed to take on biomass, then removed and tested in a sep tank for minor ammonia digestion testing, maybe 2 weeks, that fast.
sponge placed inside clean new tank, using fishless cycle technique dr tims/any of them (all the same) about 2-3 weeks, just like fishless cycling for dry rock tanks
all are estimates, the final say in any cycling is when 1 ppm or so can be digested in a body of water within 24 hours via the substrates in question, in this case a sponge. the whole thing is skewed unless very high quality ammonia test kits designed to read low levels accurately are used. since your qt tank will use this as its primary source of nitrification you need a digest test to know when its ready. add liquid cleaners ammonia or dr times Achloride to .5-1ppm verified accurate on the best test kit search returns show do well indicating low readings, wait 24 hours test again. if zero, ready.