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](https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-35/issue-2/Limit-Theorems-for-the-Maximum-Term-in-Stationary-Sequences/10.1214/aoms/1177703551.full)

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