We report on a zero-field ${}^{63}\mathrm{Cu}$ nuclear quadrupole resonance (NQR) study of nonmagnetic Mg impurity substituted ${\mathrm{Cu}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{GeO}}_{3}$ (single crystals; the spin-Peierls transition temperature ${T}_{\mathrm{sp}}\ensuremath{\sim}14,$ 13.5, and 11 K for $x=0,$ 0.0043, and 0.020) in a temperature range from 4.2 to 250 K. We found that below ${T}^{*}\ensuremath{\sim}77 \mathrm{K},$ Cu NQR spectra are broadened and nonexponential Cu nuclear spin-lattice relaxation increases for undoped and more remarkably for Mg-doped samples. The results indicate that random lattice distortion and impurity-induced spins appear below ${T}^{*},$ which we associate with a precursor of the spin-Peierls transition. Conventional magnetic critical slowing down does not appear down to 4.2 K below ${T}_{\mathrm{sp}}.$