In this paper, we provided various estimates of the first nonzero eigenvalue of the $\beta$-Laplacian operator on closed orientated $p$-dimensional slant submanifolds of a $2m$-dimensional complex space form ${\tilde{\mathbb{V}}}^{2m}(4ϵ)$ with constant holomorphic sectional curvature $4ϵ$. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $\beta$-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.