Optical nonlinearities can be strongly enhanced by operating in the so-called near-zero-index (NZI) regime, where the real part of the refractive index of the system under investigation approaches zero. Here we experimentally demonstrate semi-degenerate four-wave mixing (FWM) in aluminum zinc oxide thin films generating radiation tunable in the visible spectral region, where the material is highly transparent. To this end, we employed an intense pump (787 nm) and a seed tunable in the NIR window (1100-1500 nm) to generate a visible idler wave (530-620 nm). Experiments show enhancement of the frequency conversion efficiency with a maximum of 2% and a signal-to-pump detuning of 360 nm. Effective idler wavelength tuning has also been demonstrated by operating on the temporal delay between the pump and signal.