Calcium can be added to cheese milk to influence the coagulation process and to increase cheese yield. Calcium compounds used in the dairy industry show substantial differences in their practical application. Therefore, this study aimed to evaluate the potential use of 0, 5, 10, 15, and 20 mg Ca 100 g−1 of milk in the form of calcium gluconate, lactate, and carbonate as alternatives to calcium chloride in manufacturing fresh acid rennet cheese from high-pasteurized (90 °C, 15 s) goat’s milk. The pH value of the cheese was reduced most strongly by the addition of increasing doses of calcium lactate (r = −0.9521). Each cheese sample showed increased fat content with the addition of calcium. Only calcium chloride did not reduce protein retention from goat’s milk to cheese. The addition of 20 mg Ca 100 g−1 of milk in the form of gluconate increased cheese yield by 4.04%, and lactate reduced cheese yield by 2.3%. Adding each calcium compound to goat’s milk significantly increased Ca and P levels in the cheese (p ≤ 0.05). The highest Ca levels were found in cheese with the addition of 20 mg Ca 100 g−1 of milk in the form of lactate. In all groups, similar contents of Mn, Mo, and Se were found. Calcium addition significantly affected cheese hardness, while higher calcium concentrations increased hardness. Carbonate caused the greatest increase in the cohesiveness of cheese. The addition of calcium compounds increased the adhesiveness and springiness of cheese compared to controls. The cheese with calcium chloride had the highest overall acceptability compared to the other cheese samples. The addition of calcium carbonate resulted in a lower score for appearance and consistency, and influenced a slightly perceptible graininess, sandiness, and stickiness in its consistency, as well as provided a slightly perceptible chalky taste.