Non-Hermitian (NH) systems with aperiodic order display phase transitions that are beyond the paradigm of Hermitian physics. This motivates the search for exactly solvable models, where localization-delocalization phase transitions, mobility edges in complex plane, and their topological nature can be unraveled. Here, we present an exactly solvable model of quasicrystal, which is a nonpertrurbative NH extension of a famous integrable model of quantum chaos proposed by Grempel et al.[Phys. Rev. Lett. 49, 833 (1982)] and dubbed the Maryland model. Contrary to the Hermitian Maryland model, its NH extension shows a richer scenario, with a localization-delocalization phase transition via topological mobility edges in complex energy plane.