In this paper, we investigate Fibonacci polynomials as complex hyperbolic functions. We examine the roots of these polynomials. Also, we give some exciting identities about images of the roots of Fibonacci polynomials under another member of the Fibonacci polynomials class. Finally, we obtain some excellent relationships between the roots of Fibonacci polynomials and the modular group, Hecke groups and generalized Hecke groups with geometric interpretations.