Well, it looks like no one will guess, so I'll go ahead and tell you. The year was 1990. That was the last year, according to my research, that all starting SEC quarterbacks were white. From 1991 to 1993, there was a black QB at Kentucky named Pookie Jones, and every year since 93 there has been at least one black QB in the SEC.
Remember, the SEC split into divisions in 1992, after the admission of Arkansas and South Carolina that year. Arkansas had a black QB in 1990 named Quinn Grovey, who is part of their radio broadcast team now. South Carolina had a white QB in 1990 named Bobby Fuller. So if they had been in the SEC in 1990, then we would have to go further back than that. I'll post the 1990 SEC QB's names later on.