N = 2 supersymmetric Yang-Mills theory with flavor hypermultiplets at finite temperature is studied for finite quark number density (nb) by a dual supergravity background with nontrivial dilaton and axion. The quarks and their number density nb are introduced by embeddings a probe D7 brane. We find a critical value of the chemical potential at the limit of nb = 0, and it coincides with the effective quark mass for nb = 0. At this point, a transition of the D7 embedding configurations occurs between their two typical ones. The phase diagrams of this transition are shown in the plane of chemical potential versus temperature for Yang-Mills theory at finite temperature. In this phase transition, the order parameter is considered as nb. This result seems to be reasonable since this theory is in the quark deconfiment phase.