Regulating the physical properties such as the quantum phase and the Kondo effect of molecular electronic devices near critical points may play a key role in increasing the robustness of quantum memory, which is a crucial component in quantum information processing. Molecules with a triangular topology are ideal prototypes to reveal the competition among magnetic frustration, Kondo screening, and local inter-molecule exchange interactions. Herein, motivated by a recent work investigating the single-electron tunneling through a redox-active edge-fused porphyrin trimer by using a Hubbard dimer model [J. O. Thomas, J. K. Sowa, B. Limburg, X. Bian, C. Evangeli, J. L. Swett, S. Tewari, J. Baugh, G. C. Schatz, G. A. D. Briggs, H. L. Anderson and J. A. Mol, Chem. Sci., 2021, 12, 11121], we studied the phase transition, the electronic transport, and the thermodynamical properties of a real molecular trimer structure organized in a triangular topology, with and without an external magnetic field, and at zero and non-zero temperatures. Both the Hubbard electron-electron interaction and the Heisenberg exchange interaction are fully taken into account, with the aid of the state-of-the-art numerical renormalization group method. Various kinds of Kondo behaviors and quantum phase transitions are demonstrated, due to the competition among the Ruderman-Kittel-Kasuya-Yosida interaction, the direct exchange coupling, and the Zeeman effect. Our findings may offer deep insights into the manipulation of the quantum phase and the Kondo behavior in a molecular trimer with a triangular topology.