Well, I know when you first start the engine cold, there is an electric motor with an air pump attached that runs, it's part of the pollution equipment, called the AIR pump. It runs for pretty close to 2 minutes, so try starting it cold, listen and see if it absolutely STOPS at about 2 minutes. This is a "stone cold" engine, after sitting overnight. After shutting off, after waiting about a minute, you'll also hear the motor run again for just a second, that's the computer checking the system to make sure it's functioning OK.
My bet is that it's done it all along and you just noticed it now. Sounds kind of like a vacuum cleaner noise.