Here are some fundamental ideas that hold true for citing AI generated content, no matter which citation style you're using:
Credit: Brown University Library

Whether you are going to use GENAI for course assignments, academic work, or other forms of writing You should always check with your instructor before using AI in any coursework.
Copyrights, Professional Perspective - Copyright Chaos: Legal Implications of Generative AI
Copyright and Artificial Intelligence - U.S. Copyright Office
Additional Information: Fair Use: Training Generative AI
Any content created by artificial intelligence should be carefully evaluated. Artificial intelligence can produce false or biased results and use outdated information. Use the CRAAP test to help you evaluate content generated by AI. At a minimum, ask yourself the following questions:
Credit: CRAAP test - Benedictine University Library

APA recommends citing artificial intelligence bot communications as an algorithm output. Credit the author of the algorithm. In the case of ChatGPT, the creator is the company OpenAI. The title should be the name of the AI tool. Indicate which version you are using after the title of the AI tool in brackets. Include the phrase "Large language model" in square brackets to indicate this is an AI tool. Finally, include the link to the tool. APA also recommends considering including the full text of the interaction in an appendix. See the full explanation on the APA Style's blog, How to Cite ChatGPT.
OpenAI. (2023). ChatGPT (Mar 14 version) [Large language model]. https://chat.openai.com/chat
(OpenAI, 2023)
(OpenAI, 2023; see Appendix A for the full transcript).
Omit the author. Use a description of the content in place of a title. List the AI tool and the version, as well as the company behind the tool. You may need to use a third-party browser extension, like ShareGPT to create a stable URL to the content. For more information, see the MLA Style Center's blog post, How do I cite generative AI in MLA style?
“Describe the ethics of using AI to write homework assignments” prompt. ChatGPT, Feb. 13 version, OpenAI, 22 Mar. 2023, https://sharechatgpt.com/share/8ceff295698fd9e78cebf7a92c79e231
("Describe the ethics of using AI")
The Chicago Manual Style - Online
Chicago style requires that you cite AI-generated content in your work by including either a note or a parenthetical citation, but advises you not to include that source in your bibliography or reference list. The reason given for this is that, because you cannot provide a link to the conversation or session with the AI tool, you should tread that content as you would a phone call or private conversation. However, AI tools are starting to introduce functionality that does allow a user to generate a sharable link to a chat conversation, so this guidance from the Chicago Manual of Style may change.
Here are some general guidelines for referencing AI-generated content in Chicago style:
Format:
1. Author, Title, Publisher, Date, url for the tool.
Example (if information about the prompt has been included within the text of your paper):
1. Text generated by ChatGPT, OpenAI, March 7, 2023, https://chat.openai.com/chat.
Example (including information about the prompt):
1. ChatGPT, response to "Explain how to make pizza dough from common household ingredients," OpenAI, March 7, 2023, https://chat.openai.com/chat.