Mehrotra and Özevin (SIAM J Optim 19:1846–1880, 2009) computationally found that a weighted barrier decomposition algorithm for two-stage stochastic conic programs achieves significantly superior performance when compared to standard barrier decomposition algorithms existing in the literature. Inspired by this motivation, Mehrotra and Özevin (SIAM J Optim 20:2474–2486, 2010) theoretically analyzed the iteration complexity for a decomposition algorithm based on the weighted logarithmic barrier function for two-stage stochastic linear optimization with discrete support. In this paper, we extend the aforementioned theoretical paper and its self-concordance analysis from the polyhedral case to the semidefinite case and analyze the iteration complexity for a weighted logarithmic barrier decomposition algorithm for two-stage stochastic convex quadratic SDP with discrete support.