If the correlations decay fast enough $\sigma_{ij}(n) = o(1/\log n)$, then the asymptotic distribution of the maximum is the same as if the variables were independent (i.e. the standard Gumbel distribution) - see:
Limit Theorems for the Maximum Term in Stationary Sequences, S.M. Berman (Ann. Math. Statist. 1964) Link
and also: On the asymptotic joint distribution of the sum and maximum of stationary normal random variables H.C. Ho and T. Hsing (Journal of applied probability, 1996). https://www.jstor.org/stable/3215271
For the general case (correlations decay slower or not at all) I don't know of exact results for the limit, but there is a work showing how to compute bounds on the expectation for finite $n$:
Useful Bounds on the Expected Maximum of Correlated Normal Variables, A.M. Ross (2003) https://emunix.emich.edu/~aross15/q/papers/bounds_Emax.pdf
