Failure mode and effects analysis (FMEA) helps to identify the weak points in the processing, manufacturing, and assembly of products and plays an important role in improving product reliability. To address the shortcomings of the existing FMEA methods in terms of the uncertainty treatment of information and not considering the weights and correlations between risk factors, we propose a new FMEA method. In this paper, the Fermatean fuzzy Z-number (FFZN) is proposed by fusing the Fermatean fuzzy number and Z-number. Extending it to the Bonferroni mean (BM) operator, the Fermatean fuzzy Z-number-weighted Bonferroni mean (FFZWBM) operator is proposed. A new FMEA method is proposed based on this operator. In order to overcome the factors not considered in the FMEA method, two new risk factors are proposed and added. The ability of experts to express fuzzy information is enhanced by introducing the FFS. The weights and correlations between the influencing factors can be handled by aggregating the evaluation information using the FFZWBM operator. Finally, the proposed method is applied to an arithmetic example and the accuracy of the proposed method is proved by teaming it with other methods.