J.K. Rowling clarified in an interview that Quirrell did not previously hold the Defense Against the Dark Arts position before Harry's first year:
Incidentally, I know some have asked about Quirrell with regard to this question [about the Defense Against the Dark Arts position being cursed]. He was teaching at Hogwarts for more than a year, but NOT in the post of D.A.D.A. teacher. He was previously Muggle Studies professor.
As for who was in the position before Quirrell, the short answer is that the individual has never been identified. There have only been nine DADA professors identified by name. Six were present during Harry's six years at Hogwarts (Quirrell, Lockhart, Lupin, Crouch, Umbridge, Snape), Alastor Moody accepted the position before being impersonated by Bartemius Crouch Jr., Amycus Carrow (a Death Eater) was installed during the time of The Deathly Hallows, and Galatea Merrythought was the teacher who retired while Tom Riddle was in school.
While J.K. Rowling alluded to a multi-year professor after the events of the story (thanks to the curse being lifted by the defeat of Voldemort), she never gave him or her a name. On the other end of the timeline, Rowling has never discussed, much less named, the professor before Quirrell.
So unless Rowling changes her mind and gives the individual a name, we will never know who the Defense Against the Dark Arts professor was before Quirrell. I suspect that it's not something she has ever considered herself, as the person served before Harry arrived at Hogwarts and she never intended that professor to have a legacy that would impact Harry or the others at Hogwarts.