When there is a quote within a regular quote, single quote marks are used. I assume this is done to separate them from the double quote marks around the outside quote.
But what about a block quote that doesn’t have the double quote marks around it. Should quotes inside a block quote use double or single quotes?
I believe you use double quotation marks. A quick look at the online Chicago Manual of Style confirms this:
Although material set off as a block quotation is not enclosed in quotation marks, quoted matter within the block quotation is enclosed in double quotation marks (CMS 11.35).
Note that this if from another source that quotes it, since the online CMS is paywalled.