Likely a faulty coolant temperature sensor. Change the sensor.
when the temperature light is green means that the engine is cold and needs to be warmed up.
NO it does not. The check engine light comes on when the ECU detects a problem with the emissions system and has nothing whatsoever to do with the outside temperature.
Check Engine light is on. Engine OK when cold, but when reaches operating temperature it stalls at idle.
Cold light
The blue light on the dashboard of the Infiniti Q45 typically indicates that the engine is running in a cold state and is not yet at optimal operating temperature. This light serves as a reminder for the driver to allow the engine to warm up before driving aggressively. It can also indicate that the engine's coolant temperature is low, which is normal when starting the vehicle.
engine coolant
If it comes on it means the temperature of your engine is too hot. Your engine is overheating.
Light is due to electron transitions in atoms.
Because your engine is too hot. or the light is broken.
temperature sensor is bad
because if the light changed them so would the temperature
because if the light changed them so would the temperature