It's mostly the nature of the beast. Mine's done it since new.

Some have had good results replacing the motor mounts.
IIRC there was one post that claimed the crankshaft pulley was the issue.

 

My 2010 Patriot does this. I was told my a mechanic that it was the Constant Velocity Transmission. I took it to a locally owned transmission shop and as soon as I told the owner the make and model of the car he started laughing. He told me that with 130,000 miles on it and it being a CVT that the shaking/vibrating was inherent with these CVT's. He explained that at that mileage the metal belt of the CVT had stretched and was vibrating. He also stated the CVT was as tough as an army tank and in the $3500--$4000 range to swap out for a rebuilt unit IF he could find one.

 

Even though they look fine, over time the mounts can still harden and compress allowing more vibration to transmit to the chassis.

 

My Patriot did this. For years. I took it to multiple shops who all told me that the mounts were fine. Finally I had enough and replaced them myself. The did look perfectly fine. As soon as I replaced them the shaking was gone.

 

1. Continuously Variable Transmission, not Constant Velocity Transmission.

2. Maybe it's your spark plugs? First time I replaced mine, it'd shake when stopped and in drive, and even quit once or twice in that situation. Gapped the plugs properly and it worked fine.

 

All automatic transmission will vibrate some in Drive and not in Neutral or Park. As Sandstone said, it's the nature of the beast. This is because the engine is at its weakest point when idling, and it is straining to turn the transmission and move the vehicle against the brakes.

Rosso makes a good suggestion. Patriot plugs are only 30,000 mile plugs so they are easily neglected. Easy DIY. There are threads on here and on Youtube about changing them if you're insecure about doing it yourself.

 

My 2017 Pat (Hyundai 6 speed transmission) vibrates at stop lights, but not when the transmission is in park or neutral. Plugs are new, but motor mounts are original. Perhaps it's time Rock Auto gets more of my money, lol.