In this paper we generalize the result of James Pachares  on the distribution of a definite quadratic form to the case of non-central normal variates.
"Distribution of a Definite Quadratic Form for Non-Central Normal Variates." Ann. Math. Statist. 32 (3) 883 - 887, September, 1961. https://doi.org/10.1214/aoms/1177704981