The answer to this question is yes, calcium supplements can provoke stomach acid production, thereby aggravating symptoms of indigestion or dyspepsia for which calcium-salts are often taken in the first place.
Calcium ions act as co-stimulators of acid production from parietal and oxyntic cells in the stomach lining. These cells use a K+ / H+ ATPase to pump hydrogen ions (acid) into the stomach. The cells are switched on by a combination of histamine, acetyl choline (from vagus nerve efferents supplying the stomach wall) and calcium ions.
Therefore, taking calcium carbonate (chalk) indigestion tablets, or drinking milk (which is rich in calcium) can cause temporary relief of gastrointestinal symptoms, because the calcium carbonate reacts with the stomach acid to form carbon dioxide and water.
But when this calcium is subsequently absorbed it can provoke rebound hyper-acidity, making the initial problem worse. In other words an uncomfortable vicious circle.