Quantum Yang-Baxter equation has been proposed by Baxter in the context of a particular model of statistical mechanics (6-vertex model ??) and called star-triangle relation. Later it has been generalized and axiomatized to a number of contexts: it is most notably satisfied by the universal R-element in a quasitriangular Hopf algebra. In some context it is equivalent to a braid relation for certain transposed matrix. The solution to a quantum Yang-Baxter equation for matrices is called the quantum Yang-Baxter matrix or quantum R-matrix; some solutions to quantum Yang-Baxter equation have good limits in classical mechanics which are classical r-matrices, and the latter satisfy the classical Yang-Baxter equation.
A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univ. Press 1994