At present, under the conditions of climate change, for mainly environmental but also economic reasons, especially in the case of new wheat genotypes, alternative cropping systems are recommended in addition to the common conventional system. The aim of this study was to determine the effect of the integrated system (INTEG) and conventional system (CONV) on the physiological parameters, yield, and mineral composition of the grain, as well as the amount and quality of protein of winter wheat Hymalaya (hybrid cv.) and Formacja (population cv.) against the background of changing hydrothermal conditions in the years of the study. The field experiment was carried out in 2016–2019 in Przecław (50°11′00″ N, 21°29′00″ E), Poland. More favorable values of physiological parameters and grain yield were found in the CONV system than in the INTEG system. A more efficient course of the photosynthesis process in cv. Hymalaya effected a higher grain yield, which was similar in the INTEG system to that of cv. Formacja from the CONV system. The use of the CONV system effected an increase in the grain quality traits as well as the sum of gliadins and glutenins, including the subunits γ gliadins, LMW glutenins, and HMW glutenins. Grain of cv. Hymalaya from the INTEG system had higher contents of Fe, Mn, and Mg and more favorable composition of glutenin proteins and their HMW/LMW ratio than cv. Formacja. Higher values of quality traits and gluten protein fractions and subunits, along with a reduction in the grain yield of wheat cultivars, were favored by periods with rainfall deficit in the wheat ripening period, where low hydrothermal coefficients were recorded.