The stability of milk proteins is affected by changes in the pH value of milk, the heating temperature, and the addition of calcium compounds or chelating agents, which can cause alterations in calcium distribution. The purpose of this study was to determine the potential of the use of calcium citrate to manufacture fresh acid rennet cheese from high-temperature-pasteurized goat’s milk (90 °C, 15 s) from the spring and autumn season and the effect of the calcium dose used on the physicochemical and organoleptic properties of the cheese. Autumn milk was found to be a richer source of total solids, confirming the effect of the production season on milk quality. The applied doses of calcium did not cause the denaturation of goat milk proteins and allowed pasteurization to take place at 90 °C for 15 s. The addition of calcium citrate resulted in a significant increase in the pH value of milk and cheese compared to the control sample. Adding 15 and 20 mg of Ca 100 g−1 to milk as citrate had the most beneficial effect on increasing protein retention in cheese in both seasons, showing a rise from 1.33% to 2.40%. The production season significantly influenced the cheese yield. The control goat cheese from the autumn season showed a 6.85% higher yield compared to the spring cheese. An increase in cheese yield was also observed as the calcium dose of milk increased. The content of micro- and microelements in cheese was affected by the production season. The addition of calcium citrate to milk resulted in a significant increase in the calcium content of cheese—from 120.83 to 147.45 mg 100 g−1 in the spring season and from 130.66 to 151.21 mg 100 g−1 in the autumn season. Increasing the dose of calcium increased the hardness of cheese samples by 1.37 N in the spring and 0.90 N in the autumn. The organoleptic evaluation showed that adding calcium to milk did not significantly affect the organoleptic characteristics of goat cheese.