MgZnO/ZnO quantum wells on top of ZnO nanowires were grown by pulsed laser deposition. Ensembles of spatially fluctuating and narrow cathodoluminescence peaks with single widths down to 1 meV were found at the spectral position of the quantum well emission at 4 K. In addition, the number of these narrow QW peaks increases with increasing excitation power in micro-photoluminescence, thus pointing to quantum-dot-like emission centers. Indeed, laterally strained areas of about 5 nm diameter were identified at the quantum well positions on top of the nanowires by high-resolution transmission electron microscopy.