Not really...
...it just looks that way because the light off takes place pretty quick...it only takes about 3-4 seconds to get the N1 up to 12% where you can introduce fuel (with the condition lever), then it lights off a second or two later, and you look up and the prop's turning.
However...if you just motored the starter (you use a switch position called STARTER ONLY, which just turns the engine but doesn't energize the igniters), you'd push enough air through the gas producer to "blow" the power turbine wheel around...thus turning the prop. The prop wouldn't turn for a few seconds until the motored air flow was sufficient to overcome the inertia of the PT wheel/driveshaft/gearbox/prop combination. This is why it appears that the prop only turns after lightoff in a normal start.
Obviously the prop would turn immediately if you motored a geared engine such as a Garrett on an MU-2, or the Allison on the C-130...