  We consider a numerical approximation of a linear quadratic control problem
constrained by the stochastic heat equation with non-homogeneous Neumann
boundary conditions. This involves a combination of distributed and boundary
control, as well as both distributed and boundary noise. We apply the finite
element method for the spatial discretization and the linear implicit Euler
method for the temporal discretization. Due to the low regularity induced by
the boundary noise, convergence orders above 1/2 in space and 1/4 in time
cannot be expected. We prove such optimal convergence orders for our full
discretization when the distributed noise and the initial condition are
sufficiently smooth. Under less smooth conditions, the convergence order is
further decreased. Our results only assume that the related (deterministic)
differential Riccati equation can be approximated with a certain convergence
order, which is easy to achieve in practice. We confirm these theoretical
results through a numerical experiment in a two dimensional domain.
