The Laplace-Beltrami operator on a sphere with a cut arises when considering the problem of wave scattering by a quarter-plane. Recent methods developed for sound-soft (Dirichlet) and sound-hard (Neumann) quarter-planes rely on an a priori knowledge of the spectrum of the Laplace-Beltrami operator. In this paper we consider this spectral problem for more general boundary conditions, including Dirichlet, Neumann, real and complex impedance, where the value of the impedance varies like $\alpha /=r,\; r$ being the distance from the vertex of the quarter-plane and α being constant, and any combination of these. We analyse the corresponding eigenvalues of the Laplace-Beltrami operator, both theoretically and numerically. We show in particular that when the operator stops being self-adjoint, its eigenvalues are complex and are contained within a sector of the complex plane, for which we provide analytical bounds. Moreover, for impedance of small enough modulus |α|, the complex eigenvalues approach the real eigenvalues of the Neumann case.