$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is 
$$\sum_{k,l}A_{k,l}\,EY_iY_jY_kY_l,$$
where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the [Isserlis theorem][1]: 
$$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}.$$

  [1]: https://en.wikipedia.org/wiki/Isserlis%27_theorem#Even_case