Should the link include the quotes?

I guess it's not material. You can include double quotes in the anchor text. Or you can exclude them together with the exclamation mark :)

This is not a question about language, or right and wrong, but about what looks better on a web page. That will depend on various factors. By default, HTML underlines all the characters in linked text. That means that if the quote marks are included within the link, they will be underlined, and I think this can look ugly because, of course, unlike the letters themselves, the quote marks sit well above the line. Also, it may look odd to have punctuation marks underlined because in the offline world underlining usually indicates emphasis, and we emphasize words, not their punctuation.

However, many web sites, including this one, use CSS to change the default style for links, and it is quite popular just to use a color that is different from the unlinked text, without any underlining. In such a case I can see no problem with including the quote marks within the link, so that they are the same color as the text they are enclosing. I think it probably looks slightly better that way, but it is really a matter of taste.