How close to Nice would these have to be? I'm assuming fairly close since you mentioned where in Nice you're staying. Would the Cotes du Rhone region be too far gone? It's a 3-4 hour drive to Avingon from Nice, but that region is world-renowned for its wine.
As for something perhaps within the city limits (or not far beyond), here are a couple:
Château de Cremat
Château de Bellet
I believe these both conduct tours if you let them know you're coming. Nice isn't really known for its wine prowess though, so I can't vouch too much for either.