"They have lived in what is now Ghana for many centuries," she went on. Does this sentence need quotations

Yes. You've written the sentence correctly with quotation marks in the right places.

You are good as is

No, this sentence does not need quotations. Quotations are typically used to indicate directly spoken words or to highlight a specific phrase or expression. In this case, the sentence is written in a narrative or reported speech format, with the phrase "she went on" indicating that someone is continuing to speak. This sentence is not a direct quote, but rather a narration of what someone said or stated.