If I am using an accelerometer to externally wake up my microcontroller, will my accelerometer consume power all the time? or only during a event detection?
it depends on accelerometer but in the common case YES, the accelerometer will work! Almost all accelerometers allow setup threshold to generate an interrupt, in this case, you can reduce consumptions by the set minimal sampling frequency. And some accelerometers have the internal mechanism to detect primitive motions, consumption in this also can be various.
And don't forget to provide independent power supply to accelerometer