We report the $^{121/123}$Sb-NMR/nuclear quadrupole resonance (NQR) measurements on the newly-discovered superconductor BaTi$_2$Sb$_2$O with a two-dimensional Ti$_2$O square-net layer formed with Ti$^{3+}$ (3$d^1$). NQR measurements revealed that the in-plane four-fold symmetry is broken at the Sb site below $T_{rm A} sim$ 40 K, without an internal field appearing at the Sb site. These exclude a spin-density wave (SDW)/ charge density wave (CDW) ordering with incommensurate correlations, but can be understood with the commensurate CDW ordering at $T_{rm A}$. The spin-lattice relaxation rate $1/T_1$, measured at the four-fold symmetry breaking site, decreases below superconducting (SC) transition temperature $T_{rm c}$, indicative of the microscopic coexistence of superconductivity and the CDW/SDW phase below $T_{rm A}$. Furthermore, $1/T_1$ of $^{121}$Sb-NQR shows a coherence peak just below $T_{rm c}$ and decreases exponentially at low temperatures. These results are in sharp contrast with those in cuprate and iron-based superconductors, and strongly suggest that its SC symmetry is classified to an ordinary s-wave state.
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