Google Search Console shows the user's query if the query is popular enough and your website is in the search results. Bing shows all queries, even if they are not popular, and if your website is in the search results.
But if AI recommends your website when answering people's questions, you cannot find out what questions the user discussed, how many times your website was shown, and in what position. You can see the UTM tag in your website analytics (for example, GPT adds utm source), but that is the maximum amount of information that will be available to you. But if a user discussed a question with AI and only got your brand name, and then found your site in a search engine, you won't be able to tell that they found you with the help of AI advice.