Can an accelerometer function at a constant velocity?
Under normal conditions, an accelerometer will measure 1g acceleration from gravity moving at constant velocity (more-or-less) along the surface of the earth. It feels like it is accelerating upwards, and that is indistinguishable from actually accelerating upwards from the pov of the accelerometer.
Acceleration, by definition, is the rate of change of velocity (the derivative wrt time).
Integrating acceleration can give some indication of velocity, but accelerometers drift and unless the acceleration is fairly quick you won't really be able to distinguish that drift from actual acceleration.
You can, with an accelerometer, detect jarring and quick motions such as the initial acceleration of a car from a parked position. Of course there may be innocent shocks and so on (tapping on glass, earthquakes etc.) that could give false positives.
You could use a magnetometer to detect changes in orientation wrt the earth's magnetic field (may not be reliable in some locations). You could use GPS to detect change in position (may not be possible if the signal cannot be received).