It's possible to do add a call into your userSetup.mel or userSetup.py to write to a text or csv log file when a Maya session starts, and add a scriptJob listening to the quitApplication event that will log when a session closes. But there are some ways that system could fail -- like if Maya crashed, or if you ran multiple sessions at the same time.
A more resilient solution would be to use one of the ways that the operating system itself can log when a process is running. On windows for instance, you can set a policy to log events whenever a process starts and finishes, then use the Event Viewer to read that information. Similar things are possible on Mac and linux. With that level of tracking, you will always have the information even if you forget to log when you start or finish.
