We study dynamical quantum phase transitions in a 2-qubit system interacting with a transverse field and a quantized bosonic environment in the context of open quantum systems. By applying the stochastic Schrödinger equation approach, the model with a spin-boson type of coupling can be solved numerically. It is observed that the dynamics of the rate function of the Loschmidt echo in a 2-qubit system within a finite size of Hilbert space exhibit nonanalyticity when the direction of the transverse field coupled to the system is under a sudden quench. Moreover, we demonstrate that the memory time of the environment and the coupling strength between the system and the transverse field can jointly impact the dynamics of the rate function. We also supply a semi-classical explanation to bridge the dynamical quantum phase transitions in many-body systems and the non-Markovian dynamics of open quantum systems.