Previous insertion-based and paraphrase-based backdoors have achieved great success in attack efficacy, but they ignore the text quality and semantic consistency between poisoned and clean texts. Although recent studies introduce LLMs to generate poisoned texts and improve the stealthiness, semantic consistency, and text quality, their hand-crafted prompts rely on expert experiences, facing significant challenges in prompt adaptability and attack performance after defenses. In this paper, we propose a novel backdoor attack based on adaptive optimization mechanism of black-box large language models (BadApex), which leverages a black-box LLM to generate poisoned text through a refined prompt. Specifically, an Adaptive Optimization Mechanism is designed to refine an initial prompt iteratively using the generation and modification agents. The generation agent generates the poisoned text based on the initial prompt. Then the modification agent evaluates the quality of the poisoned text and refines a new prompt. After several iterations of the above process, the refined prompt is used to generate poisoned texts through LLMs. We conduct extensive experiments on three dataset with six backdoor attacks and two defenses. Extensive experimental results demonstrate that BadApex significantly outperforms state-of-the-art attacks. It improves prompt adaptability, semantic consistency, and text quality. Furthermore, when two defense methods are applied, the average attack success rate (ASR) still up to 96.75%.