Is the correct interpretation that you are using the KeyFobDemo project and trying to modify the behaviour of default accelerometer demo app to get your use case to work?
Do you wany anything to be transmitted OTA also?
Here is the wiki for the KeyFobDemo Accelerometer:
http://processors.wiki.ti.com/index.php/Category:KeyFobDemo#Accelerometers
The accelerometer read event (KFD_ACCEL_READ_EVT) is set to default 50ms. Maybe you can start a counter in this event to count every time it is read, and check if it was a change or not since the previous reading. If no change continue to count up til 1min, and setup a new event that can trigger the buzzer in your KeyFobApp_ProcessEvent loop.
LPRF Rocks the World