Yes you can add your custom marker to show user current location....but you need to calculate the latitude and longitude your self using locationListener and show that location on map as follows:
public void setCurretnLocMarker(loc) {
LatLng mLatLng=new LatLng(loc.getLatitude(), loc.getLongitude());
getMap().animateCamera(CameraUpdateFactory.newLatLng(mLatLng));
getMap().addMarker(new MarkerOptions().position(mLatLng).title(mLatLng.toString()).draggable(false));
}