In this paper, we study a stochastic parabolic problem involving a nonlocal diffusion operator associated with nonlocal Robin-type boundary conditions. The stochastic dynamics under consideration are driven by a mixture of a classical Brownian and a fractional Brownian motions with Hurst index $H\in(\frac{1}{2}, 1).$ We first establish local in existence result of the considered model and then explore conditions under which the resulting SPDE exhibits finite-time quenching. Using the probability distribution of perpetual integral functional of Brownian motion as well as tail estimates of fractional Brownian motion we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. The existence of global in time solutions is also investigated and as a consequence a lower estimate of the quenching time is also derived. Our analytical results demonstrate the non-trivial impact of the considered noise on the dynamics of the system. Next, a connection of a special case of the examined model is drawn in the context of MEMS technology. Finally, a numerical investigation of the considered model for a fractional Laplacian diffusion and Dirichlet-type boundary conditions is delivered.