Can I use a 1" stem on a 1 & 1/8" steer tube?
I've done the opposite (but drop bars before and after), and have a quill stem adaptor to use a non-quill stem with a threaded headset. My fork is 1", but the adaptor came with a similar shim for 1 1/8" steerers. I've also used quill stems with threadless headsets, but only with matching sizes.

I don't see any reason the product you link to won't do the job it's designed for though.

To replace a non-quill stem with a quill stem you'll need something like 
http://problemsolversbike.com/products/locking_headset_spacers, cf. "Threadless Without Spacers" at http://sheldonbrown.com/handsup.html#threadless

I assume you are replacing a 1 1/8" quill stem on your trek though?