Q. How do I cite a quote within a quote?

You will need to give credit to both the original source and the source you actually used. For example, if you read Chan's book (published in 2002) and wanted to quote or paraphrase Chan's quotation of Nguyen (who published her work in 1999), your in-text citation will look like this:

"Quote" (Nguyen, 1999, as cited in Chan, 2002, p. 487)

or

Nguyen argues, "Quote" (as cited in Chan, 2002, p.487)

On your references page, you will only list the source you actually read: in this case, Chan.

Also, from page 292 of the APA Manual:

"Enclose direct quotations within a block quotation in double quotation marks. In a quotation in running text that is already enclosed in double quotation marks, use single quotation marks to enclose quoted material."