The theory of q-rung orthopair fuzzy sets (q-ROFSs) proposed by Yager can more effectively describe fuzzy information in the real-world. Because q-ROFSs contain a parameter q and can adjust the range of expressed fuzzy information, they are more superior to intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). Archimedean T-norm and T-conorm (ATT) is an important tool to generate operational rules, so many general operational rules can be provided by ATT based on the q-rung orthopair fuzzy numbers (q-ROFNs). Meanwhile, the Bonferroni mean (BM) operator has the prominent advantage that it can consider the interrelationships between the different attributes. Therefore, it is an important and meaningful innovation to extend the BM operator to the q-ROFNs based on the ATT. In this paper, we firstly discuss and give some q-rung orthopair fuzzy operational rules by using ATT. Further we extend BM operator to the q-ROFNs, and propose the q-Rung Orthopair fuzzy Archimedean BM (q-ROFABM) operator and the q-Rung Orthopair fuzzy weighted Archimedean BM (q-ROFWABM) operator and study their desirable properties. Then, a new multiple-attribute decision-making (MADM) method is developed based on q-ROFWABM operator and its process is presented in detail. Finally, we take advantage of the practical example to verify its effectiveness and superiority by comparing with other existing methods.