Let d be a square-free integer and Z[√d] a quadratic ring of integers. For a given n ∈ Z[√d], a set of m non-zero distinct elements in Z[√ d] is called a Diophantine D(n)-m-tuple (or simply D(n)-m-tuple) in Z[√ d] if product of any two of them plus n is a square in Z[√d]. Assume that d ≡ 2 (mod 4) is a positive integer such that x2 − dy2 = −1 and x2 − dy2 = 6 are solvable in integers. In this paper, we prove the existence of infinitely many D(n)-quadruples in Z[√d] for n = 4m + 4k√d with m, k ∈ Z satisfying m ≢ 5 (mod 6) and k ≢ 3 (mod 6). Moreover, we prove the same for n = (4m + 2) + 4k√ d when either m ≢ 9 (mod 12) and k ≢ 3 (mod 6), or m ≢ 0 (mod 12) and k ≢ 0 (mod 6). At the end, some examples supporting the existence of quadruples in Z[√ d] with the property D(n) for the above exceptional n’s are provided for d = 10.