We consider the problem of quickest change detection (QCD) for a signal which may undergo both a nuisance and a critical change. Our goal is to detect the critical change without raising a false alarm over the nuisance change. An optimal sequential change detection procedure is proposed for the Bayesian formulation of our QCD problem. A sequential change detection procedure based on the generalized likelihood ratio test (GLRT) statistic is also proposed for the non-Bayesian formulation. We show that our proposed test statistics can be computed efficiently via respective recursive update schemes. We compare our proposed stopping rules with the naive 2-stage procedures, which attempt to detect the changes using separate optimal stopping procedures (i.e., the Shiryaev procedure in the Bayesian formulation, and the CuSum procedure in the non-Bayesian formulation) for the nuisance and critical changes. Simulations demonstrate that our proposed rules outperform the 2-stage procedures.