Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called generalization Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, use the method, proposed by us, we obtain the exact number of spanning trees in the Farey graph. The result shows that the computation complexity is O(n), which is better than that of the matrix tree theorem with O(n²), where n is the number of steps. We derive a basic property of generalization Farey graph controlled by a parameter k, which is closely to the number of spanning trees of network, which become bigger with increasing k. We also obtain the maximum and minimum numbers of spanning trees of these kinds of networks through the method.