Recent advances in natural language processing and machine learning have led to the development of chatbot models, such as ChatGPT, that can engage in conversational dialogue with human users. However, the ability of these models to generate toxic or harmful responses during a non-toxic multi-turn conversation remains an open research question. Existing research focuses on single-turn sentence testing, while we find that 82\% of the individual non-toxic sentences that elicit toxic behaviors in a conversation are considered safe by existing tools. In this paper, we design a new attack, \toxicbot, by fine-tuning a chatbot to engage in conversation with a target open-domain chatbot. The chatbot is fine-tuned with a collection of crafted conversation sequences. Particularly, each conversation begins with a sentence from a crafted prompt sentences dataset. Our extensive evaluation shows that open-domain chatbot models can be triggered to generate toxic responses in a multi-turn conversation. In the best scenario, \toxicbot achieves a 67\% activation rate. The conversation sequences in the fine-tuning stage help trigger the toxicity in a conversation, which allows the attack to bypass two defense methods. Our findings suggest that further research is needed to address chatbot toxicity in a dynamic interactive environment. The proposed \toxicbot can be used by both industry and researchers to develop methods for detecting and mitigating toxic responses in conversational dialogue and improve the robustness of chatbots for end users.