In this work, we consider the convergence rates of numerical methods for solving stochastic time-fractional partial differential equations in a convex polygon/polyhedron. For this model, both the time-fractional derivative and the stochastic process result in low regularity of the solution. Hence, the numerical approximation of such problems and the corresponding numerical analysis are very challenging. In our work, the stochastic time-fractional PDE is discretized by a backward Euler convolution quadrature in time with piecewise continuous linear finite element method in space for which a sharp-order convergence is established in multidimensional spatial domains with nonsmooth initial data. Numerical results are presented to illustrate the theoretical analysis.