The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$α$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $α$, namely we first assume $α$ to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when $α$ is fixed. A preparatory analysis of uniform estimates in both $α$ and discretization parameters is carried out. Starting out from the stochastic LANS-$α$ model, we achieve convergence toward the continuous strong solutions of the stochastic Navier-Stokes equations in 2D when $α$ vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-$α$ model is accomplished if $α$ is fixed.