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I have a table with county polygon boundaries (named "counties"). I need to build a query that identifies a particular county as well as any adjoining counties that share the same border. Have tried breaking this down but I'm missing something. Any ideas?

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You can create a Virtual Layer that will query the polygon layer twice and apply the desired filtering.

go to the menu Layer > Add Layer > Add/Edit Virtual Layer... and enter the query

SELECT b.*
FROM myLayer a
JOIN myLayer b ON ST_INTERSECTS(a.geometry,b.geometry)
WHERE a.id <> b.id 
AND a.id = 1234

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