Ive owed a G35 coupe for 4 years so I decided to stay within the same company and got a 13 g37x sedan 48kmiles. The car drives fine. These past few days we've been having really cold weather. After a cold start and warming up, the car is perfectly fine. Once I hit a traffic light and apply brakes, a noise can be heard on the driver side of the engine. Sounds like a deep rumble, nothing too crazy or shakey but enough to notice somethings up. Also the rpms are dropped to 500. The rpms dropping has been happening ever since i got the car and only drops when in D. I have read other threads regarding this issue. Fixing it with the dealer raising one users rpm upto 700 at idle. The manual states the idle should be 650 +/-. This issue didnt bother me too much as i was going to tackle that later on but now, after a cold start, rumbling or shaky vibrations can be heard more along the drivers side as i apply the brakes. This is at a stop light on drive. on park everything is fine. Once foot is off the brakes the rumbling stops, and continues if i apply the brakes again (mind you all while on drive). I literally applied brakes on and off moving inch by inch to see if it rumbles only when i pressed on the brakes. after a little driving, rumble goes away. Maybe will appear for a second or 2. Anyone else have this issue? I'll attach videos soon. if all else fails ill just have the dealership look at it.

 

The 500rpm shake is a known behavior on these cars. Plenty of megathreads (with no resolutions) on this site about it...
However, because this only happens when you're braking, complete speculation here, it sounds like a vaccuum leak on/around the brake booster. It must be a pinhole that closes when it's all warmed up. If you can, graph out your short and long-term fuel trims (3 samples: cold idle, highway drive, warm idle) in torque/odbfusion and see if you're running slightly lean in the cold.