Did you not read the end of the article you linked? His wife definitely had black roots, but it was a family secret. One that they might not have even known about.
Earlier this year, the mystery of whether Vivian was descended in part from Africans was finally resolved. In a February episode of the PBS show, “Finding Your Roots,” host and historian Henry Louis Gates Jr. presented Rosanne Cash with her DNA results and family genealogy. Vivian Cash’s maternal great-great grandmother was indeed an enslaved Black woman, Sarah Shields, whose White father in 1848 had granted her and her eight siblings their freedom and their passage into Whiteness, too.
Basically Vivian's great-great-grandmother was a black enslaved woman, and her descendents hid this fact to save themselves from Jim Crow laws.
It's possible she and Johnny knew but kept it quiet because they lived in the deep south in a time when it was scary to be in any way black. The 'one-drop rule' is still a thing for a lot of americans, after all. We know that Vivian wife was living in fear of the KKK whenever he went on tour. I would imagine he said whatever he had to say to keep her safe while he was away