I'm not sure about the codeshare, but it is true that discounted business class seats on Qatar have a seat selection fee. Expedia should have really highlighted this during the booking process (another reason to book direct with the airlines), but I am presuming it is complex as likely you/they don't know the underlying fare class when booking a codeshare.

I would honestly say there is no such thing as a bad QSuites seat, so I'm not sure if its worth the money - but seems to be in your case. In this case it seems likely you should pay the fee and then complain to Expedia that it wasn't disclosed at booking point, but I wouldn't have that much hope you'd get the money back.

Note, here is the announcement about fares being unbundled, though there have been numerous changes since, but you can see discounted business class fares have a seat fee:
https://www.qatarairways.com/content...klet_Trade.pdf
https://onemileatatime.com/news/qata...s-class-fares/