This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation 4 (an−14yn−1) + (vn − 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∑ n∈N exp ( nδ ) (|1 − an| + |vn|) < ∞ for > 0 and 1/2 ≤ δ ≤ 1