To find new spin‐gapless semiconductors (SGSs) or half‐metallic materials to promote the development of spintronics, the stable, electronic, magnetic, and mechanical properties of quaternary Heusler alloys CoX′CrZ (X′ = Sc, Ti; Z = Al, Ga) are studied by first principles. Findings show the ground state structure of all alloys is the T‐II structure, and their equilibrium lattice constants are CoScCrAl (6.15 Å), CoScCrGa (6.14 Å), CoTiCrAl (5.94 Å), and CoTiCrGa (5.93 Å), respectively, indicating that the T‐II structure is easily synthesized. The new SGSs of CoScCrAl and CoScCrGa have an indirect bandgap at the Fermi level in spin‐down channels. The indirect bandgap values are 0.52 and 0.56 eV, respectively, and the origin of the bandgap is analyzed. The spin polarization of all alloys is more than 90%, and CoScCrAl and CoScCrGa are suitable for room temperature applications due to their high Curie temperature and strict compliance with the Slater–Pauling rule. CoTiCrAl and CoTiCrGa conform to the Slater–Pauling rule when the lattice constant reaches 6.30 Å, and the half‐metallicity can be improved by doping or strain. All alloys have unique mechanical properties and are mechanically stable. Moreover, CoScCrAl and CoScCrGa exhibit brittle behavior, whereas CoTiCrAl and CoTiCrGa exhibit ductility.