We study superconformal indices of 4d compactifications of the 6d minimal (D_N+3, D_N+3) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with A_N, C_N and (A₁)^N global symmetries. We mainly concentrate on C₂ case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our C₂ operator and previously derived A₂ generalization of van Diejen model. Finally, we also consider compactifications with A_N-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.