Are there any exercises or natural ways to relieve TMJ pain?

Yes. There are exercises and techniques that can be utilized to help alleviate the pain associated with TMJ pain. Trigger point therapy and myofascial release are some techniques used by healthcare providers to treat pain associated with the muscles affecting the tmj.
Relax your jaw. Unless there has been some type of trauma to the face or jaw (like getting hit or punched, or falling on your chin), or if there is arthritis in the jaw joint, most TMJ pain is due to grinding or clenching your teeth. The best way to relieve the pain is to not grind or clench your teeth. Unless you are chewing food, your teeth should not be touching. Check your dentist about a night guard.