Summary: Let denote by \(F_{k,n}\) the \(n\)th \(k\)-Fibonacci number where \(F_{k,n}=kF_{k,n-1}+F_{k,n-2}\) for \(n\geq 2\) with initial conditions \(F_{k,0} = 0\), \(F_{k,1}=1\), we may derive a functional equation \(f(k, x) = kf(k, x - 1) + f(k, x - 2)\). In this paper, we solve this equation and prove its Hyers-Ulam stability in the class of functions \(f : \mathbb{N}\times \mathbb{R}\to X\), where \(X\) is a real Banach space.