Beneficial nonlinearities have been intentionally introduced to improve energy harvesting performance. However, few results are reported on the issues of exploiting bistable and quasi-zero stiffness structures or mechanisms simultaneously to enhance ocean energy conversion in the literature. In this study, a novel bistable electromagnetic wave energy converter is proposed, which includes a bio-inspired X-shaped supporting structure/mechanism and a mechanical-motion-rectifier. This is for the first time that the bistable stiffness and quasi-zero stiffness are employed together (in all working range) to strengthen wave energy conversion, by developing a novel bistable X-structured system as a wave energy converter, which can bear with large vibration stroke simultaneously. Additionally, a novel mechanical-motion-rectifier with a novel energy-storing-releasing mechanism is specially designed as the power take-off system to convert linear motion to rotary motion, which greatly improves the instant voltage and power generated. The novel X-structured system demonstrates very designable nonlinear bistable and quasi-zero stiffness and thus can significantly enhance the wave energy conversion performance. Compared with traditional linear converters, much more power can be generated, especially in the low-frequency range due to the quasi-zero and bistable properties. The experiments show that, with the wave height of 0.15 m as well as the wave period of 4 s, it can generate a voltage peak at around 15 V and a power peak at about 1 W. The present results indicate that, the proposed nonlinear bistable X-structured system holds promising potentials in acting as a floating-point energy converter, which could power seashore environmental monitoring systems and wireless communication devices for smart oceans.