Composite sol-gel materials of the MgO–Al2O3–ZrO2–SiO2 system were synthesized and the processes of their thermal evolution and crystallization were studied. Application of sol-gel compositions of the MgO–Al2O3–ZrO2–SiO2 system to the surface of quartz ceramics leads to a significant increase in the mechanical strength of the material. The processes of thermal evolution of the sol-gel composition were studied using IR spectroscopy, X-ray diffraction, electron microscopy, and energy dispersive analysis. It was shown that the formation of the oxide composite structure of materials begins at the stage of wet gels. Treatment of quartz ceramics with composite sols followed by drying and heat treatment up to 1200°C results in modification of the surface layers of the material, which makes it possible to increase the mechanical strength of the material by more than 20%. Sol-gel modifying compositions, upon drying and subsequent heat treatment, form polycrystalline structures bonded to quartz ceramic particles and consisting of various oxide crystals.