While synthesis methods for pure ZnO nanostructures are well established, an efficient technique for the growth of ZnO-based nanowires or microrods that incorporate any type of quantum structure is yet to be established. Here, we report on the fabrication and optical properties of axial Zn1-xMgxO/ZnO/Zn1-xMgxO quantum wells that were deposited by molecular beam epitaxy on ZnO microrods obtained using a hydrothermal method. Using the emission energy results found in cathodoluminescence measurements and the results of a numerical modeling process, we found the quantum well width to be 4 nm, as intended, at the growth stage. The emission of quantum well-confined excitons persists up to room temperature. We used the fabricated structures to determine the carrier diffusion length (>280 nm) in ZnO using spatially resolved cathodoluminescence. The micro-photoluminescence results suggest an increase in the electron-phonon coupling strength with increasing microrod size.