We consider a class of Volterra equations with 3-monotone kernels driven by additive Gaussian noise in infinite dimensions. We discretize the
spatial variable using a standard finite element method. The time derivative is approximated by the Backward Euler method and the convolution is discretized using Lubich's convolution quadrature based on the Backward Euler scheme. Precise convergence rates are obtained, depending on the regularity of the noise, in the mean squared norm (strong convergence). We also establish weak convergence rates for twice differentiable test functions. It turns out that the weak convergence rate is twice the rate of strong convergence.