Often, lower back "muscle pain" is referred pain from a herniated disc; especially given the high prevalent of discogenic pain found in our middle-aged American population. As far as I know, there are no specific studies that prove that coughing activates the quadratus lumborum or multifidi muscles. In fact, coughing activates other repiratory muscles in the chest and abdominal muscles. So it's hard to attribute coughing to muscle strain. However, coughing does increase the abdominal pressure and that pressure can increase intradiscal pressures. If a protruding disc is present, coughing can elicit pain.

Multifidi spasms or strains may be secondary to the underlying inflammation of the dorsal rami at the lumbar disc level. It's important to be evaluated by a back specialist like a physiatrist to distinguish if the pain is actually from the muscle or is referred from irritation of the nerves to the back from a herniated disc. Without a good diagnosis, it's been shown that patient's outcomes with medications and therapy are less effective. If the pain is found to be from a disc, the treatment is similar to a muscle strain. But there are much more effective treatments available as well.