The properties of the color-flavor-locked (CFL) quark matter under strong magnetic fields at finite temperatures within a quasiparticle model are investigated in this work. Our results indicate that CFL quark matter pressure becomes anisotropic under strong magnetic fields, while its equation of state (EOS) and equivalent quark mass are both strongly affected by temperature, energy gap constant Δ, and strong magnetic field inside the CFL quark matter. The equivalent quark mass of CFL quark matter decreases with temperature and magnetic field strength increasing, which implies an inverse magnetic catalysis phenomenon. The results also indicate that the entropy per baryon of the CFL quark matter increases with temperature rising and decreases with Δ increasing. Furthermore, the properties of CFL magnetars in different isentropic stages are studied. The star mass and radius depend primarily on the strength and orientation of magnetic fieldinside the CFL magnestars. The maximum star mass increases with entropy per baryon increasing, while the star matter temperature rises at high isentropic stage. Moreover, the polytropic index of the CFL quark matter decreases with star mass increasing.