In this paper, we propose an energy harvesting-based overlay hybrid satellite-terrestrial spectrum sharing system which exploits both direct and relay links from a primary satellite source to terrestrial user with the coexistence of a secondary transmitter-receiver pair on the ground. It is assumed that an energy-constrained secondary transmitter deploys the time switching-based relaying protocol to harvest energy from the primary satellite's signal and then performs an amplify-and-forward based relay cooperation with the primary satellite network to explore an opportunity for spectrum access. By adopting Shadowed-Rician fading for satellite links and Nakagami-m fading for terrestrial links, we derive analytical expressions for the outage probability of primary satellite network and secondary terrestrial network. In addition, to get useful insights, expressions for throughput and energy efficiency of primary satellite network are also provided assuming a delay-limited scenario. Moreover, our results illustrate the impact of power splitting factor and energy harvesting time fraction on the system performance.