The Modular group Γ is the most well-known discrete group with many applications. This work investigates some subgraphs of the subgroup Γ 3 , defined by (a b c d) ∈ Γ : ab + cd ≡ 0 (mod 3) In [1], the subgraph F 1,1 of the subgroup Γ 3 ⊂ Γ is studied, and Fibonacci numbers are obtained by means of the subgraph of F 1,1. In this paper, we give a generalization of the subgraphs generating Fibonacci numbers for the subgroup Γ 3 and some subgraphs having special conditions