You can evaluate the EGBMG function using the double Mellin-Barness Integral.
Check this book for the "two variables Meijer G function which is a special case of the H-function:
N. T. Hai and S. B. Yakubovich, The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory, World Scientific, Series on Soviet and East European Mathematics, Vol.6},
isbn = {981-02-0690-9}, 1992,
You can also a Matlab implementation for special cases in :
H. Chergui and M. Benjillali and S. Saoudi, IEEE Commun. Lett., Performance Analysis of Project-and-Forward Relaying in Mixed {MIMO}-Pinhole and {Rayleigh} Dual-Hop Channel,2016
