The ¹³⁷Ba NMR and NQR spectra as well as the spin-lattice relaxation rate were measured for the first time in the BaPb_xBi_1-xO₃ samples with x=0.91(normal metal), x=0.75 (superconductor) and x=0.64 (semiconductor). The strong quadrupole broadening of the NMR lines with lowering temperature was found for all the samples. An evolution of static distortions in the BiO₆-octahedra sublattice along with asymmetry of the electric field gradient at the Ba-sites is discussed. This asymmetry caused by the nearest non-cubic surrounding of barium is originated from the temperature dependent tilt of BiO₆-octahedra. Spin-lattice relaxation rate of ¹³⁷Ba is monitored by fluctuating part of electric field gradient at the Ba-site and its exponential growth at low temperatures evidences the thermally activated collective rotations of BiO₆-octahedra around their static tilt. An activation energy and a strength of these dynamic distortions are evaluated and found to increase in going from metallic to semiconducting phase of BaPb_xBi_1-xO₃.