Immobile. The liver is nicely fixed in place under the right diaphragm. It would take quite a hit to move or displace it - you would be in an emergency room if that happened. So your quivering feeling is from something else in that region of the body: stomach distress, abdominal muscle contration, etc. If bothersome or persistent, check it out with your family doctor.
Liver movement? No. Your liver is quite firmly fixed (it's quite a heavy organ) to the right upper side of your abdominal cavity. It can enlarge in certain diseases (alcoholism, heart failure, metabolic storage diseases, infiltrative diseases, or cancer, etc.), and this is experienced by feeling the lower edge of the liver expand downward out from under the lower rib margins.