Characterizing nonequilibrium dynamics in quantum many-body systems is a challenging frontier of physics. In this Letter, we systematically construct solvable nonintegrable quantum circuits that exhibit exact hidden Markovian subsystem dynamics. This feature thus enables accurately calculating local observables for arbitrary evolution time. Utilizing the influence matrix method, we show that the influence of the time-evolved global system on a finite subsystem can be analytically described by sequential, time-local quantum channels acting on the subsystem with an ancilla of finite Hilbert space dimension. The realization of exact hidden Markovian property is facilitated by a solvable condition on the underlying two-site gates in the quantum circuit. We further present several concrete examples with varying local Hilbert space dimensions to demonstrate our approach.