The influence of randomly quenched disorder in the incommensurate phases of Rb₂(Zn_1-xCu_x)Cl₄ (for x = 0.03), Rb₂(Zn_1-xCd_x)Cl₄ (for x = 0.03 and 0.05), Rb₂(Zn_1-xHg_x)Cl₄ (for x = 0.03 and 0.05) and Rb₂Zn(Cl_1-xBr_x)₄ (for x = 0.01 and 0.03) is investigated via the amplitudon and phason dynamics using ³⁵Cl nuclear quadrupole resonance studies. Defect pinning at the metal and halogen sites in the prototype compound Rb₂ZnCl₄ has been attempted for the first time and has yielded novel results. Quenched randomness at the metal site (Zn) in Rb₂ZnCl₄ induced strong pinning of the modulation wave (irrespective of the size of the dopant compared to the host). This is evident from a temperature-independent and consequently T₁ unlike the case for impurity pinning at the other sites (cation and anion). The effect is enhanced with increasing concentration of the dopant. This result is contrasted with defect pinning at the halogen site (Cl) in Rb₂ZnCl₄ with Br substitution which induced weak pinning of the modulation wave (temperature-dependent and consequently T₁) similarly to substitution at the cation site as seen from earlier studies. Furthermore, the impurities have been categorized as random-field or random-potential type by evaluating the symmetry parameter (m) associated with the impurity. It is seen that Cu, Cd and Hg are random-field-type impurities inducing strong pinning of the modulation wave (m < 6; m = 6 for Rb₂ZnCl₄) while the Br impurity is of a random-potential type inducing a weak pinning of the modulation wave.