A polymeric multilayered mirror doped with a saturable dye worked as a passive Q-switch of a laser-diode-pumped ${\mathrm{Nd}}^{3+}:\mathrm{Y}\mathrm{V}{\mathrm{O}}_4$ microchip laser. The multilayered mirror consisted of alternately spin-coated layers of polyvinylcarbazole (PVK) and cellulose acetate on a glass substrate. A dye of bis-(dimethylaminodithiobenzil)-nickel was doped in one of the layers of PVK, providing the repetitively Q-switched pulses. The pulse width and repetition rate were $4\mathrm{ns}$ and $68\mathrm{kHz}$ , respectively, for a laser-cavity length of $5\mathrm{mm}$ , and the average and the peak power were $40\mathrm{mW}$ and $156\mathrm{W}$ , respectively, for the highest pump power of $435\mathrm{mW}$ . The dependence of the Q-switched characteristics on the pump power and on the concentration of the doped dye is described.