Stock selection for effective investment decisions is a valuable and attractive research interest for many years. Owing to the uncertainty and complexity of the stock market, many fuzzy multicriteria decision-making (MCDM) methods were proposed to solve stock selection problems. However, these methods have difficulty in characterizing unreliable information, which is widespread in the stock market, and handling the non-compensation among multiple criteria. In this paper, an innovative method is developed from the perspectives of information reliability and criterion non-compensation to manage stock selection problems. First, the Z-number, which is a powerful tool for describing real-life information and identifying information reliability, is introduced to depict stock evaluation information. Second, the outranking degree of Z-numbers is defined based on the fuzzy and probability information. Subsequently, some outranking aggregation and exploitation procedures are presented based on the idea of Elimination and Choice Translating Reality (ELECTRE) I to handle the non-compensation among stock evaluation criteria. By integrating the above studies, a Z-number ELECTRE I MCDM method is developed. Finally, a stock investment object selection problem is solved, and some discussions and analyses are conducted to testify the applicability and validity of this method.