In an AI search, four things happen: the question becomes one or more search queries, a handful of documents is retrieved from an index, the model writes the answer from them and shows some as visible sources.
Your website can enter the retrieval step if it is crawlable and relevant to the question, and the sources if the model uses it. It does not enter the answer by magic: it enters because it was among what was retrieved.
Without retrieval, the model answers from what it learned in training, and then it cites nothing.