Just to clarify, usually when Alkalinity is referred to in the reef hobby, it can mean two related but slightly different things. Alkalinity generally means water's capacity to neutralize an acid. So as this might relate to pH, it would be the water's resistance to acid causing the pH to drop. When we test for alkalinity with our test, what we are actually doing is testing this capacity.
However, most people equate Alkalinity with Carbonate Alkalinity, which is a measure of the the presence of carbonate and bicarbonate ions (this does not include calcium!). So people often list their Alkalinity in dKH which is specifically carbonate alkalinity. Reefers care a lot about carbonate alkalinity, because along with calcium, corals require carbonate to form their calcium carbonate skeleton. This is what reefers usually care about more than the waters ability to neutralize acid.
In order to test calcium levels for coral you'd also need a separate test kit, and this would not be tested by an Alkalinity test kit.
pH can also be important to test. Chronic low pH (7.8 or lower) can interfere with coral calcification. However, generally you're better off testing for Alkalinity and Calcium to start with. Keeping those levels in place, usually keeps your pH in place as well.