I checked out LonelyPlanet.com at the following link:
http://www.lonelyplanet.com/italy/sh..._order=popular
The second one listed seemed the most relevant, Bassetti Tessuti in Rome.
ETA link:
http://quiltinggallery.com/quilting-bloggers/c/105/
Quilting blogs in Italy - you may poke around on these and find some ideas for shops,etc.