to remind you of what the split polygon tool is and what it does go to the tutorials that come with maya
then navigate through:
getting started with maya > polygon modeling > lesson 1: Modeling a polygonal mesh > splitting polygon faces

and that's the split polygon tool in use

alternatively you can use the insert edge loop tool as hammer suggested
it's much quicky when you want to split a strip of quads up (it only works with faces taht clearly have an opposite edge)

as for your NURBS conversion, the only thing i can think of is that your tesselation method is by count and you don't have the same isoparams on both sides of the surface