This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation Δ(an-1Δyn-1) + (νn-λ)2 yn = 0, n ∈ ℕ and the boundary condition (γ0 + γ1λ) y1 + (β0 + β1λ) y0 = 0 where (an); (vn) are complex sequences, γi, βi ∈ ℂ, i = 0, 1 and λ is a eigenparameter. The paper presents Jost solution, eigenvalues, spectral singularities and states some theorems concerning quantitative properties of the spectrum of this BVP under the condition {formula presented}.