We describe the correspondence of the Matsuo-Cherednik type between the quantum n-body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N |M ). The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the Z 2 -grading for a fixed value of N + M , so that N + M + 1 different qKZ systems of equations lead to the same n-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantumclassical correspondence between the classical n-body Ruijsenaars-Schneider model and the supersymmetric GL(N |M ) quantum spin chains on n sites.where the number of indices j k such that j k = a is equal to M a for all a = 1, . . . , K. The dual vectors J are defined in so that J J ′ = δ J,J ′ .Then the statement of the qKZ-Ruijsenaars correspondence is as follows [19]. For any solution of the qKZ equations (1.1) Φ = J Φ J J from the weight subspace V({M a }) the function Ψ = J Φ J , Φ J = Φ J (x 1 , ..., x n ) (1.7) 4 The quantum R-matrices entering (1.2) are assumed to be unitary: R ij (x)R ji (−x) = id. 5 The set {e ab | a, b = 1...K} is the standard basis in Mat(K, C): (e ab ) ij = δ ia δ jb .