Q: Would a faulty alternator cause smoking from under the hood on the passenger side?

If your alternator is failing, this can definitely cause it to smoke depending on how badly it has failed. When an alternator fails, the bearings inside usually have failed to a point that causes them to whine severely and, depending on how badly they have failed, they may also smoke due to the excessive friction they create. I would recommend having an expert from YourMechanic come to your location to diagnose and replace your alternator.