We propose logarithmic-barrier decomposition-based interior-point algorithms for solving two-stage stochastic quadratically constrained convex quadratic programming problems in a Hilbert space. We prove the polynomial complexity of the proposed algorithms, and show that this complexity is independent on the choice of the Hilbert space, and hence it coincides with the best-known complexity estimates in the finite-dimensional case. We also apply our results on a concrete example from the stochastic control theory.