As of today (and much discussion is going on so keep aware of any changes) use the one cell quotation marks (dots 236, and dots 356) for any double quotation marks used in a book if there is one type of print double quotation marks. It doesn't matter if they are slanted or straight quotes...as long as they are all the same. According to Braille Formats Appendix G, these symbols do not need to be listed on the Special Symbols page.
If you also have single quotes in the same book, use the single braille quotes (dots 6, 236 and dots 6, 356). These symbols are not required on the Special Symbols page (although it is noted in Appendix G that the closing single quote is new and MAY be listed- and if the closing single quote is listed, I would also list the opening single quote for consistency).
If the book has more than one type of double quote marks and they are used for different things, then the two-cell symbols for quotes should be used to represent one of the two types...and a note would be required so the reader knows which braille symbol represents which type of quote.
I realize that 7.6.1 of UEB says that the nonspecific quotation marks are required on the special symbols page or in a TN, but Braille Formats does not require these symbols to be listed unless there is some reason to explain the print form.
As for what description to use ... if you are going to go with 7.6.1 and list the nonspecific quotation marks, then you must follow the rest of 7.6.1 and state what the print form is as part of your description (are they straight quotes, curly quotes, etc). And to be honest, if only one type of double quote is used, I think it really only matters that they are quotes...making it unnecessary to put them on the Special Symbols page as the "regular" braille double quotes have been used for a long time and are likely known to the braille reader already.
As I said earlier, I do know that quotation marks continue to be a subject of discussion on the ICEB code maintenance committee listserve - and they are working to clarify the symbols and the usage for them.