The short answer is that it depends on the language on the insurance policy. There are many different types of polices and coverage. It is always best to review the exact policy language rather than make assumptions.
Most business insurance does not cover claims for breach of contract. However, some customer complaints might be covered by errors & omission policies, or something similar. Much also depends on the type of injury the customer has. For example, if they are claiming some sort of physical or property damage, then coverage is more likely.