Zero- and longitudinal-field muon-spin-relaxation (ZF- and LF-$\ensuremath{\mu}$SR) measurements were carried out on the impurity-doped spin gap systems TlCu${}_{1\ensuremath{-}x}$Mg${}_{x}$Cl${}_{3}$. In a slightly doped case of $x=$ 0.0047, no evidence for a static internal magnetic field was observed down to 20 mK although the specific heat indicated the magnetic phase transition at $T$ $=$ $0.70$ K as previously reported [J. Phys. Soc. Jpn. 78, 074705 (2009)]. Above $x=$ 0.006, the existence of a static internal magnetic field is confirmed by LF-$\ensuremath{\mu}$SR, and with increasing Mg concentration of $x$, the internal magnetic field, and the volume fraction of a spin frozen region where the static internal magnetic field appears increase simultaneously. These results suggest that the magnetic ground state changes from the spin singlet state to a spin fluctuating state, and to a spin frozen state by the impurity doping, and also suggest there exist a threshold doping ratio in the appearance of a static staggered moment in the $S=1/2$ dimer systems weakly coupled by three-dimensional interactions.