Due to the contribution of photon tunneling, the output power of near-field thermophotovoltaic (NFTPV) devices can surpass that of far-field thermophotovoltaic devices by several orders of magnitude, which has recently drawn extensive attention. Previous research explores the enhancement brought by hyperbolic materials, but little investigation has been done on the frequency shift of hyperbolic bands. Hyperbolic metamaterials acquire the hyperbolic properties through complex nanofabrication processes. However, calcite (${\mathrm{Ca}\mathrm{CO}}_{3}$), as a natural hyperbolic material, possesses hyperbolic properties without the need for expensive processing. Despite this, ${\mathrm{Ca}\mathrm{CO}}_{3}$ is rarely chosen as the thermal emitter in NFTPV devices. In this work, we propose an NFTPV device using ${\mathrm{Ca}\mathrm{CO}}_{3}$ as the thermal emitter and an $\mathrm{In}\mathrm{Sb}$ p-n junction as the PV cell. Furthermore, we compare the performance of four structures to analyze the effect of the graphene layer on the ${\mathrm{Ca}\mathrm{CO}}_{3}$-$\mathrm{In}\mathrm{Sb}$ NFTPV devices we propose. The numerical results demonstrate that the ${\mathrm{Ca}\mathrm{CO}}_{3}$/$\mathrm{graphene}/\mathrm{In}\mathrm{Sb}$ structure exhibits the best performance based on the assumption of considering only radiative recombination. The ${\mathrm{Ca}\mathrm{CO}}_{3}$/$\mathrm{graphene}/\mathrm{In}\mathrm{Sb}$ structure simultaneously achieves 41% efficiency and 94 W/${\mathrm{cm}}^{2}$ output power density when the temperature of the thermal emitter is 900 K. The physical mechanism of the significantly improved performance of the NFTPV device arises from the frequency coupling between the hyperbolic resonance and the interband transition of the $\mathrm{In}\mathrm{Sb}$ p-n junction, where the surface plasmon polaritons excited in graphene make a promotion role. Moreover, we discuss the impact of the gap distance, the temperature of the thermal emitter, and the thickness of the thermal emitter on the performance of NFTPV devices. Our research contributes to understanding the coupling of surface plasmon polaritons and hyperbolic phonon polaritons and the design of high-performance thermophotovoltaic device.