In this Rapid Communication, a set of $^{209}$Bi-nuclear magnetic resonance (NMR)/nuclear quadrupole resonance (NQR) measurements has been performed to investigate the physical properties of superconducting (SC) BaTi$_2$Bi$_2$O from a microscopic point of view. The NMR and NQR spectra at 5~K can be reproduced with a non-zero in-plane anisotropic parameter $eta$, indicating the breaking of the in-plane four-fold symmetry at the Bi site without any magnetic order, i.e., `the electronic nematic state. In the SC state, the nuclear spin-lattice relaxation rate divided by temperature, $1/T_1T$, does not change even below $T_{rm c}$, while a clear SC transition was observed with a diamagnetic signal. This observation can be attributed to the strong two-dimensionality in BaTi$_2$Bi$_2$O. Comparing the NMR/NQR results among BaTi$_2$$Pn$$_2$O ($Pn$ = As, Sb, and Bi), it was found that the normal and SC properties of BaTi$_2$Bi$_2$O were considerably different from those of BaTi$_2$Sb$_2$O and BaTi$_2$As$_2$O, which might explain the two-dome structure of $T_{rm c}$ in this system.
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