FYI just in case you didn't know, it's "quote unquote", signifying the starting and ending quotation marks.
If you want to argue over details that make little difference though, I'd recommend using the term "splitting hairs" instead.
'kind unkind'
'known unknown' (unless your Donald Rumsfeld)
'seen unseen'
etc.
I feel like writing quote-on-quote takes away the ability of non-native speakers to search for and learn the basis of the idiom. I really don't see how it would be more understandable than just typing "quote".