Belching, or burping air from the mouth is not a potential direct side effect of gall bladder surgery but could be indirectly related.
Belching is caused by swallowing air, most commonly subconsciously. In other words, the person isn't aware of doing so. Some belching is common and normal, particularly after eating. However, some people will consult with a doctor when belching becomes frequent and repetitive.
The most common cause of belching is called, functional belching, with the term functional referring to a problem with how the body works. Medical tests, such as x-rays and endoscopy with a scope through the mouth into the stomach are normal and do not show a medical cause of functional symptoms. Many functional symptoms are associated with stress, anxiety, and emotional upset, which can affect the person at the subconscious level.
Sometimes belching is associated with an uncomfortable or painful disease, such as gastroesophageal reflux disease (GERD), which causes heartburn and/or chest pain, or with the upper abdominal pain associated with an ulcer.
Here are two examples of cases I have seen in which belching was the indirect result of gall bladder surgery. The most common side effect of gall bladder surgery is a change of bowel pattern with loose stools or diarrhea, particularly after eating. In the first case, the belching that developed was related to the stress of not knowing why diarrhea developed after the surgery and the symptom disappeared with explanation, reassurance, and medication that was effective for the diarrhea. In the second, the belching was associated with a painful upper digestive tract disease that had been mistakenly diagnosed as a gall bladder problem, but which gradually worsened after surgery.