Considering a single-mode laser system with cross-correlated additive colored noise and multiplicative colored noise, we study the effects of correlation among noises on the normalized intensity correlation function C(s). C(s) is derived by means of the projection operator method. The effects of the self-correlation time \tau1 of the additive colored noise, \tau2 of the multiplicative colored noise, and the effect of the cross-correlation time \tau0 between the two noises on C(s) are discussed by numerical calculation. For the case of positive correlation (\lambda>0), it is found that when a0>0 the normalized intensity correlation function C(s) increases with the increase of \tau0 or \tau2, and with value of \tau0 or \tau2 becoming larger, C(s) comes to saturation. With increasing the self-correlation time \tau1 of the additive noise, a minimum and a maximum will appear on curve of C(a) as a0>0. If a0<0, C(s) decreases with the increase of \tau0, \tau1, and \tau2.