In this paper, we aim to obtain quantum error correcting codes from codes over a non-local ring R-q = F-q + alpha F-q. We first define a Gray map phi from R-q(n) to F-q(2n) preserving the Hermitian orthogonality in R-q(n) to both the Euclidean and trace-symplectic orthogonality in F-q(2n). We characterize the structure of cyclic codes and their duals over R-q and derive the condition of existence for cyclic codes containing their duals over R-q. By making use of the Gray map phi, we obtain two classes of q-ary quantum codes. We also determine the structure of additive cyclic codes over R-p2 and give a condition for these codes to be self-orthogonal with respect to Hermitian inner product. By defining and making use of a new map delta, we construct a family of p-ary quantum codes.