The theory of error-correcting codes is one of the most important issues of quantum computation and quantum information. One of the most useful techniques for reducing the effects of noise is the use of the quantum error correction codes. This paper presents a hyperbolic geometry approach to the construction of new quantum color codes. The families of quantum color codes are constructed based on the identification of compact surfaces by hyperbolic tessellations. Codes of these families have 4 and 6 minimum distances and their encoding rate is near to 1. We also provide some quantum color codes with minimum distance of at least eight and a comparison table of quantum codes.