We give a new idea how to consider some important properties of the hyperbolic metric to obtain results of the Schwarz-Pick type for the class of HQC mappings. By using some estimates for hyperbolic partial derivatives for HQC mappings, that act between various domains, we discus the hyperbolic bi- Lipschicity of those mappings. Key words: Quasiconformal mapping, Harmonic mapping, Riemann surface, Universal covering, Conformal metric, Gaussian curvature, Hyperbolic density, Hyperbolic length, Hyperbolic distance, Hyperbolic derivative, Quadratic diffe- rential, Quasi-isometry, Lipschitz i co-Lipschitz mapping.