Considering a gain-noise model for two-mode laser system driven by a periodic signal and two-color cross-correlation noises, we calculate the power spectrum and signal-to-noise ratio (SNR) of an output signal by means of linear approximation method, and discuss the influence of the system coefficients on SNR. Results show that under a certain condition satisfied by the coefficients of the system and input signals, the stochastic resonance phenomenon may be detected in the evolution of SNR, and also at certain intensities of noise, saturation coefficient c2, cross coupling coefficient b, and input signal frequency Ω.