We report on near-quantum-limited intensity noise of Nd:YVO4 microchip lasers pumped with pump-noise-suppressed diode lasers. The low-frequency intensity noise of the microchip lasers is found to depend on mode correlation effects in the absorbed radiation from the diode laser pump source. A minimum intensity noise of 0.5 dB above the standard quantum-noise limit at a frequency of 250 kHz is obtained by pumping with a grating-feedback diode laser. An accurate description of the measured intensity-noise spectra by a quantum-mechanical Langevin rate-equation model is achieved by consideration of the nonlinear gain saturation that is due to the finite lifetime of the lower laser level.