From the 2016 Census, the Guugu Yimidhirr language appears to be the most commonly spoken Aboriginal language in Cape York, spoken in 18.3% of households. This figure varies throughout Cape York of course, for example in Hope Vale it is spoken in 74.5% of households.
Edit: For some reason my comment disappears if I try and include links. The census information is just from the Census Quickstats website.
EWB Australia acknowledges the traditional custodians of country throughout Australia and recognise their continuing connection to land, waters and community. We pay our respects to them, their cultures and their land; to Elders both past and present; and to emerging leaders. We recognise that the Aboriginal & Torres Strait Islander people never ceded sovereignty of what we call Australia.