I would recommend ArnoldiMethods.jl, which should be more stable than the alternatives. You may read the release announcement at [ANN] ArnoldiMethod.jl v0.4 . It does handle generalized eigensystems, although you need to implement a bit of boilerplate code yourself (the documentation tells you what to do).
If your matrix has any special structure (e.g., banded), there might be alternatives.