Third-order intermodulation distortion (IM3) is an important metric for electromechanical resonators used in radio frequency signal processing applications since it characterizes the nonlinearity of the device, and the amount of in-band interference it generates when subject to unwanted, out-of-band signals. In this letter, we measure and model IM3 in a strain-engineered graphene mechanical resonator operated as a graphene resonant channel transistor (G-RCT). The device analyzed in this work has a voltage third-order intercept point (VIIP3) of 69.5 dBm V at a gate-to-source DC bias (Vgs) of 2.5 V, which drops to 52.1 dBm V at Vgs=4.5V when driven with two out-of-band input tones spaced 5 and 10MHz from the resonant frequency. The decrease in the VIIP3 with Vgs coincides with an increase in the transmission response (S21) of the device, illustrating a trade-off between transduction efficiency and linearity. In addition, we find that conventional micro-electro-mechanical systems theory for IM3 calculation does not accurately describe our measurement data. To resolve this discrepancy, we develop a model for IM3 in G-RCTs that takes into account all of the output current terms present in the embedded transistor structure, as well as an effective Duffing parameter (αeff).