An efficient passively Q-switched $\mathrm{Yb}:\mathrm{LuAG}$ microchip laser with ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ as saturable absorber was demonstrated for the first time to our knowledge. Slope efficiencies of 40% and 28% were measured for the initial transmission of ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ , $T_0=95\%$ and 90%, respectively. Laser pulses with a pulse energy of $19\mathrm{\mu }\mathrm{J}$ and a pulse width of $610\mathrm{ps}$ at the repetition rate of $12.8\mathrm{kHz}$ were achieved for $T_0=90\%$ ; the corresponding peak power of over $31\mathrm{kW}$ was obtained. The lasers oscillated at two or three longitudinal modes owing to the broad emission spectra of $\mathrm{Yb}:\mathrm{LuAG}$ and mode selection by ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ thin plate acting as an intracavity etalon.