The search for exotic superconductivity with high-transition-temperature ($T_c$) in two dimensions is a central theme in condensed matter physics and materials science. Here, we report on the experimental observation of intriguing two-dimensional superconductivity with $T_c$ of 3.8 K in the heterostructure of Mott insulator Ti$_2$O$_3$ and polar semiconductor GaN revealed by the electrical transport and magnetization measurements. Furthermore, at the verge of superconductivity we find a wide range of temperature independent resistance associated with vanishing Hall resistance, demonstrating the emergence of quantum metallic state with the Bose-metal scenario of the metallic phase. By tuning the thickness of Ti$_2$O$_3$ films, the emergence of quantum metallic state accompanies with the appearance of superconductivity as decreasing in temperature, indicating that the two-dimensional superconductivity is evolved from the quantum metallic state driven by the cooperative effects of the electron correlation and the interfacial coupling between Ti$_2$O$_3$ and polar GaN. Our work sheds new light into exploring the intriguing two-dimensional superconductivity via heterostructure engineering and promising potentials for the development of superconductor-based devices.