I can’t find in classical style guidelines if there is a defined standard for naming works “[Instrument] Quartet” or “Quartet for [Instrument]” (for example, “Flute Quartet” or “Quartet for Flute”). Currently the database contains a mix of both. Of course the same applies to “Trio”, “Quintet” etc.
The approach “Quartet for [Instrument]” looks more flexible because it easily extends to multiple instruments, for example “Quartet for Flute, Oboe and Strings”. On the other hand, for works with one non-Strings instrument the approach “[Instrument] quartet” (without mentioning Strings) appears to be pretty common, like mentioned “Flute Quartet”.