The stochastic resonance in an overdamped harmonic oscillator driven by cross-correlation Gaussian white noises in which one of the noises is modulated by a periodic signal is studied. By means of the Fourier transform of the stationary correlation function, an exact expression of the signal-to-noise ratio for the overdamped harmonic oscillator stochastic model is derived. Results reveal that there are two types of stochastic resonance in an overdamped harmonic oscillator stochastic model. One type of stochastic resonance is that a resonance peak exists in the curves of signal-to-noise ratio versus the multiplicative noise intensity Q. Another type of stochastic resonance is that a resonance peak exists in the curves of signal-to-noise ratio versus the oscillator frequency ω. By changing the value of signal frequency Ω, the signal-to-noise ratio versus Q curve can exhibit three different forms: the form with a signal peak, the form with a peak following a valley, and the monotonic form.