The Quadratic Conic Eigenvalue Complementarity Problem (QCEiCP) is investigated without assuming symmetry on the matrices defining the problem. We present a new sufficient condition for existence of solutions of QCEiCP, extending to arbitrary pointed, closed and convex cones a condition known to hold when the cone is the nonnegative orthant. We also address the Conic Eigenvalue Complementarity Problem (CEiCP) when the matrices are symmetric. We show that this symmetric CEiCP reduces to the computation of a stationary point of an appropriate merit function on a convex subset of the cone. Furthermore, we discuss the use of the so called Spectral Projected Gradient (SPG) algorithm for solving the CEiCP when the cone of interest is the Second Order Cone (SOCEiCP). A new algorithm is designed for the computation of the projections required by the SPG method to deal with the SOCEiCP. Numerical results are included to illustrate the efficiency of the SPG method and the new projection technique in practice.