The integration of high throughput satellite into Internet of Things (IoT) is regarded as an effective strategy to provide ubiquitous broadband access in a seamless, cost-efficient manner. Meanwhile, due to the demand of machine-to-machine (M2M) high throughput services, millimeter-wave (mmWave) IoT networks arouses huge interest. In this paper, we investigate the performance of an amplify-and-forward (AF) mmWave hybrid satellite-terrestrial relay networks (HSTRN) for IoT broadband communications, where we assume source-relay link undergos Shadowed-Rician fading and the relay-destination link undergos Rayleigh fading. Considering rain attenuation is the main factor at mmWave bands, we utilize the multidimensional rain attenuation model to analyze the effect of rain attenuation on system performance. Then we derive the closed-form expression of outage probability and tight approximation of ergodic capacity. Finally, numerical and simulation results are provided to validate our analytical results and show the effect of rain attenuation on the system performance.