Motivated by recent studies that reported the successful synthesis of monolayer Mg(OH)$_{2}$ [Suslu textit{et al.}, Sci. Rep. textbf{6}, 20525 (2016)] and hexagonal (textit{h}-)AlN [Tsipas textit{et al}., Appl. Phys. Lett. textbf{103}, 251605 (2013)], we investigate structural, electronic, and optical properties of vertically stacked $h$-AlN and Mg(OH)$_{2}$, through textit{ab initio} density-functional theory (DFT), many-body quasi-particle calculations within the GW approximation, and the Bethe-Salpeter equation (BSE). It is obtained that the bilayer heterostructure prefers the $AB^{prime}$ stacking having direct band gap at the $Gamma$ with Type-II band alignment in which the valance band maximum and conduction band minimum originate from different layer. Regarding the optical properties, the imaginary part of the dielectric function of the individual layers and hetero-bilayer are investigated. The hetero-bilayer possesses excitonic peaks which appear only after the construction of the hetero-bilayer. The lowest three exciton peaks are detailedly analyzed by means of band decomposed charge density and the oscillator strength. Furthermore, the wave function calculation shows that the first peak of the hetero-bilayer originates from spatially indirect exciton where the electron and hole localized at $h$-AlN and Mg(OH)$_{2}$, respectively, which is important for the light harvesting applications.
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