The separating redundancy is an important property in the analysis of the error-and-erasure decoding of a linear block code. In this work, we investigate the separating redundancy of the duals of first-order generalized Reed-Muller (GRM) codes, a class of nonbinary linear block codes that have nice algebraic properties. The dual of a first-order GRM code can be specified by two positive integers m and q and denoted by R(m,q), where q is the power of a prime number and q≠2. We determine the first separating redundancy value of R(m,q) for any m and q. We also determine the second separating redundancy values of R(m,q) for any q and m=1 and 2. For ≥3, we set up a binary integer linear programming problem, the optimum of which gives a lower bound on the second separating redundancy of R(m,q).