Dynamical phase transitions in the relaxation behavior of stochastic quantum walks are investigated, focusing on systems where coherent unitary evolution is periodically interrupted by dephasing. This interplay leads to a classicalization of the dynamics, effectively described by non-equilibrium Markovian processes that can violate detailed balance. As a result, such systems exhibit a richer and more complex spectral structure than their equilibrium counterparts. Extending recent insights from classical Markov dynamics [G. Teza et al., Phys. Rev. Lett. 130, 207103 (2023)], it is demonstrated that these quantum-classical hybrid systems can host not only first-order dynamical phase transitions – characterized by eigenvalue crossings – but also second-order transitions marked by the coalescence of eigenvalues and eigenvectors at exceptional points. Two paradigmatic models are analyzed: a quantum walk on a ring under gauge fields and a walk on a finite line with internal degrees of freedom, both exhibiting distinct mechanisms for breaking detailed balance. These findings reveal a novel class of critical behavior in open quantum systems, where decoherence-induced classicalization enables access to non-Hermitian spectral phenomena. Beyond their fundamental interest, these results offer promising implications for quantum technologies, including quantum simulation, error mitigation, and the engineering of controllable non-equilibrium quantum states.