Stochastic resonance (SR) in a second-order linear oscillator with quadratic trichotomous noise driven by an external periodic signal is investigated. By applying the property of the trichotomous noise and using the Shapiro-Loginov formula, the exact expression for the output spectral amplification (SPA) of the oscillator is obtained. Results show that the SPA varies non-monotonically with the amplitude, with the correlation rate, with the flatness of the trichotomous noise, as well as with the frequency of the driving signal. Moreover, the non-monotonic dependence of the SPA on the damping coefficient, on the frequency of the linear oscillator, as well as on the coefficient of the quadratic noise is also found.