twr welcome to the QR forum. I hope that you will find the dashboard a useful tool when looking for answers.

Surprisingly your question has not been answered yet so here is a reference to the DOH lounges http://dohahamadairport.com/relax/lounge. This reference lists the lounges at DOH and for most lounges the facilities available. Showers are conspicuous by their absence from the Oryx lounge description. This suggests that there are no showers in the lounge.