This research delves into a comprehensive investigation of a class of $\psi$-Hilfer generalized fractional nonlinear eigenvalue equation originated from a capillarity phenomenon with Dirichlet boundary conditions. The nonlinearity of the problem, in general, do not satisfies the Ambrosetti-Rabinowitz (AR) type condition. Using critical point theorem with variational approach and the $(S_{+})$ property of the operator, we establish the existence of positive solutions of our problem with respect to every positive parameter $\xi$ in appropriate fractional $\psi$-Hilfer spaces. Our main results is novel and its investigation will enhance the scope of the literature on differential equation of fractional $\psi$-Hilfer generalized capillary phenomena.