I suggested an edit for this answer which only changed the quoting for the text to be used for a web search. (The suggested edit has been accepted.) The original used parentheses for quotation; I changed this to
<code>. Using <kbd> might be more semantically correct as such is keyboard input rather than a code fragment, but the styling used by SE for <kbd> seems excessively jarring.
For ordinary quotations, quotation marks would be used and there are common methods for handling quotation nesting. However, using "search text" and 'extra "search text"' may be confusing when the search text itself does not contain any quoted text because the reader may assume the quotation is part of the search text since '"search text"' seems awkward.
Although this could probably be made a question on the main site, I am asking specifically for advice here at Writers.