Coherent phonon dynamics in condensed-phase medium are responsible for important material properties including thermal and electrical conductivities. We report a structural dynamics technique, time-resolved surface third-harmonic generation (TRSTHG) spectroscopy, to capture transient phonon propagation near the surface of polycrystalline CaF2 and amorphous borosilicate (BK7) glass. Our approach time-resolves the background-free, high-sensitivity third harmonic generation (THG) signal in between the two crossing near-IR pulses. Pronounced intensity quantum beats reveal the impulsively excited low-frequency Raman mode evolution on the femtosecond to picosecond timescale. After amplified laser irradiation, danburite-crystal-like structure units form at the glass surface. This versatile TRSTHG setup paves the way to mechanistically study and design advanced thermoelectrics and photovoltaics.