More than likely the problem is located in the cars emissions system. Ninety percent of the time the problem is the vehicles EGR valve. Carbon and varnish build up collect inside the valve not allowing it to operate.
there can be multiple things causeing a rough idle situation . Throttle position sesor , idle air control valve. If the vehicle has a MIL on have it check to detemine the problem with the vehicle. A rough idle can also come from somethin as simple as a bad air filter or O2 sesor.
The catch: cars is like humans. When it's cold it shivers. Those parts that holds the engine, and all those moving parts are frozen stiff. It requires a little shaking to make it work properly. Which also translates that the heater/heating equipment may not be in top shape. Rev it a little bit and it may solve the problem.