I grew up in County Down, and that was certainly how we referred to it. But rarely or never with "the", or the other way round (Down County). I think putting "the" before it may a southern Irish usage, but it wasn't normal in N.I.

The postal address is Co. Down, N.I. Likewise County Antrim, etc, although the other counties are more often referred to just by their names (as most English ones are). "Down" seems to need "County" because it's a word with other meanings and could be confusing when used on its own. A quick Google shows you just how prevalent the form "County Down" is.