When Kelly left Cincinnati for ND, Quinn was the interim head coach before Butch Jones took over. In the same coaching carousel, Turner Gill left Buffalo for Kansas, and Quinn replaced Gill at Buffalo.

Now clearly, Cincinnati is a better coaching job than Buffalo, so that progression appears a bit confusing. I'm not 100% sure about the details, but my best guess would be that either: (a) Quinn already had agreed to take the Buffalo job before Kelly took the ND job (although remember, Tom Herman reportedly had agreed to the LSU job before Texas offered, then reneged on LSU once he got the Texas offer); or, more likely, (b) Quinn had been told that he would not receive the permanent HC job at Cincinnati, and began looking elsewhere.

Quinn didn't do very well at Buffalo, and eventually was fired there. At that time, he probably hooked up with Kelly again. Certainly understandable, given the circumstances.