Is there any connection between bells palsy and tmj?

None. Bell's palsy is paralysis of the facial nerve. In no way related to TMJ. However, I have see patients with malignancy of the parotid gland who were misdiagnosed as "TMJ?" develop facial paralysis because the cancer invaded the facial nerve. Be careful if you have both symptoms.
Trigeminal Neuralgia. Your symptoms may be a neurological ailment called trigeminal neuralgia. Not a very well understood problem and should see your doctor, who may then possibly refer to a neurologist. There can be a TMJ relation, but tn, associated with bell's palsy, can create symptoms that mimic tmj/tmd.
Possibly. Tmj merely refers to the joint between the lower jaw and the skull. Bells palsy is a symptom of any condition that interrupts the facial nerve. The TMJ and the facial nerve are located near each other. Pain in or around that joint can have many causes. Usually however, joint pain does not affect the entire side of the face and loss of muscle control as does bells palsy.
Generally No. Bell's palsy is a sudden weakness of muscles of one side of the face of unknown origin and most often improves spontaneously. Tmj stands for temporo mandibular joint and the term is used loosely for a variety of conditions related to the joint between the lower jaw and the base of the skull and the surrounding musculature and ligaments usually involving pain and limited mouth opening.