In this paper we prove that there exist infinitely many square-free pairs of the form n^2+n+1, n^2+n+2. We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number. A key point in our proof is the application of bijective correspondence between the number of representations of number by binary quadratic form and the incongruent solutions of quadratic congruence.