Yes (the nominative of plural is "biurka", "biurek" is the genitive of plural), in singular it would be "Nie mamy wąskiego biurka" = "We don't have a narrow desk".
Why was 'desks' also given as a translation for 'wąskich'? Is 'wąska' so commonly used to refer to a biurka that you can use it on its own in Polish as an abbreviation of 'wąska biurka', or was this just a technical glitch?
Definitely a mistake ;) Fixed now.
"biurko" is neuter. It's "wąskie biurko". in singular and "wąskie biurka" in plural.
As a native English speaker, I consider that the more natural translations "We haven't any narrow desks." and "We have no narrow desks." should be accepted as correct.
Duolingo rejects both, insisting on "got": "We haven't got narrow desks." (OK as colloquial English) and "We have not got narrow desks." (unnaturally stilted).