Quantum correlations among different parts of a composite quantum system are the fundamental resource of several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, the quantum correlations in the Yang-Baxter spin-1/2 chain mode are investigated. In the second part of the paper, the Yang-Baxter spin-1/2 chain modes are constructed from the Yang-Baxter equation. First, we analyze the two matrix representations of Temperly-Lieb algebra. Second, the two solutions of the Yang-Baxter equation are generated using the Yang-Baxterization. Finally, we can change the usual two-particle spin-1/2 chain to the Yang-Baxter spin-1/2 chain modes by means of the unitary Yang-Baxter matrix-R. In the third part, the density matrices of the two chain modes are generated in the thermal equilibrium state in a canonical ensemble. According to the definition of the geometric measure of quantum discord, the analytical expressions of the geometric measure of quantum discord, in the temperature and the external magnetic field, are obtained for the Yang-Baxter spin-1/2 chain modes. When the temperature and the magnetic field intensity increase, the geometric measure of quantum discord decreases. Under the specific conditions, the result of the second chain mode is similar to that of the first one. Then we obtain the numerical results of quantum discord, the geometric measure of quantum discord, and concurrence. It is found that the concurrence can quickly decrease to the value of zero when the temperature is greater than the value of one. At the same time, quantum discord and the geometric measure of quantum discord are not of the value of zero. Thus the quantum discord and the geometric measure of quantum discord can go beyond the concept of entanglement and obtain the “quantumness” of the correlations between the two parts of a system for the Yang-Baxter spin-1/2 chain modes. They are very good quantum resources for quantum information and quantum computing.