The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let $L_{n}^{8,4}$ represents a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of $L_{n}^{8,4}$, we get the corresponding M\”{o}bius graph $MQ_{n}(8,4)$. In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of $MQ_{n}(8,4)$ can be determined by the eigenvalues of two symmetric quasi-triangular matrices $\mathcal{L}_{A}$ and $\mathcal{L}_{S}$ of order $4n$. Nextly, owning to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of $MQ_{n}(8,4)$.