In this paper, we obtain generating set of polynomials of two-dimensional cyclic codes over the ring $R=\mathbb{Z}_4[u]/\langle u^2-1\rangle$, where $u^2=1$. Moreover, we find generator polynomials for two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$ and specify a lower bound on minimum distance of free 1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$.