For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in the type A situation, for $\mathfrak{gl}_n$ or $\mathfrak{sl}_n$. The relation they get is
$$
(\hat{R} - q)(\hat{R} + q^{-1}) = 0,
$$
where $R$ is the R-matrix for the vector representation of $U_q(\mathfrak{g})$, $\hat{R} = \tau \circ R$, where $\tau$ is the tensor flip, and all the conventions are those of Klimyk and Schmudgen.

In other situations, the map $\hat{R}$ has more than just two eigenvalues, so the spectral decomposition is more complicated, and the characteristic polynomial (i.e. the Hecke relation) is more complicated.