The effects of the correlations between real and imaginary parts of the quadratic pump noise on the normalized intensity correlation function and the associated relaxation time of a single-mode laser model were investigated. The expressions of the intensity correlation function C(s) and the relaxation time Tc were derived. Based on numerical computations, we found that Tc monotonously increases as the strength λp of correlation between the real and imaginary parts of quadratic pump noise increases and the larger λp value is, the slower C(s) decays. The presence of correlations between the real and imaginary parts of the quadratic pump noise causes the intensity fluctuation to increase.