Quirrell era o professor Estudos dos Trouxas , antes de viajar. Uma vez que ele conheceu Voldemort na Albânia, ele retornou (para o primeiro ano de Harry) como professor de Defesa Contra as Artes das Trevas, (não durando mais de um ano).
Para confirmação, veja esta entrevista de JK Rowling em O Caldeirão Furado :
Has the jinx on the DADA teaching post at hogwarts been lifted?
Yes, at last! Incidentally, I know some have asked about Quirrell with regard to this question.
He was teaching at Hogwarts for more than a year, but NOT in the post of DADA teacher. He was previously Muggle Studies professor.