The q‐rung orthopair fuzzy set (^qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real‐life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval‐valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q‐rung orthopair fuzzy environment. First, the work defines two new order‐α divergence measures for ^qROPFSs to quantify the information of discrimination between two ^qROPFSs. We also examine several mathematical properties associated with order‐α ^qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order‐α ^qROPF entropy measures” to measure the degree of fuzziness associated with a ^qROPFS. We show that the proposed order‐α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision‐making approach to solve multiple attribute group decision‐making problems under the ^qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision‐making steps and effectiveness of the proposed approach.