In this paper, we study the Lie-generalized Fibonacci sequence and the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We derive several interesting properties of the Lie-Fibonacci sequence and relationship between them. We also give a couple of sufficient conditions for the existence of the integral points on the hyperbola $mathfrak{h}^a:x^2-axy+y^2=1$ and $mathfrak{h}_k:x^2-axy+y^2=-k$ ($k{in}mathbb{Z}_{>0}$). To list all the integral points on that hyperbola, we find the number of elements of ${Omega}_k$.