With a laser diode as the pump source, a passive Q switching of an intracavity frequency-doubling $\mathrm{Nd}:\mathrm{Gd}\mathrm{V}{\mathrm{O}}_4$ green laser with periodically poled KTP has been realized with a ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ saturable absorber. The dependences of the average output power, pulse width, pulse repetition rate, pulse energy, and peak power on incident pump power at different small-signal transmissions of ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ are measured. The rate equations considering the Gaussian spatial distribution of the intracavity photon density and the effects of the amplified spontaneous emission (ASE) are presented to simulate the Q-switching process of this laser, and the numerical solutions are in agreement with the experimental results. We have further discussed the influences of the ASE parameter ξ on the laser performance.