Hybrid structures incorporating both III-nitride and transition-metal dichalcogenide (TMD) semiconductors have strong application potential for light harvesting and optoelectronics. Here, we have investigated the properties of hybrid structures based on a MoSe2 monolayer coupled to an InGaN quantum well (QW). The coupling efficiency is controlled by a thin GaN barrier of variable thickness located between them. Time-integrated and time-resolved microphotoluminescence experiments show a quenching of the InGaN QW exciton emission, which increases with the decrease of the GaN barrier thickness d: the PL intensity is reduced by a factor of three for d = 1 nm as a consequence of carrier transfer to the MoSe2 monolayer. This interplay between the two semiconductors is confirmed by time-resolved photoluminescence spectroscopy, highlighting a clear reduction of the QW exciton lifetime in the presence of the monolayer. Interestingly, the coupling between the QW and the TMD monolayer is also demonstrated by measuring optically the excitonic transport properties in the quantum well: the exciton diffusion length decreases in the presence of the MoSe2 monolayer. Measuring dependences as a function of temperature highlights the role played by localization effects in the QW. All these results can be effectively interpreted by a type-II band alignment between the InGaN QW and the MoSe2 monolayer and a tunneling process between the two semiconductors.