There used to be quite a few BA/QR codeshares and there has been discussion about them before here:
Destinations bookable as BA Holiday using Qatar
It seems that at the moment, the codeshares are not there, whether this is temporary or permanent is an open question. See here for a bit of discussion regarding that:
BA cancellation of Qatar codeshare... rebooked into underlying Qatar flight
Edit: The ones that I can find:
- BA5807/QR27 DOH-MAN
- BA5808/QR28 MAN-DOH
- BA6190/QR638 DOH-DAC
- BA6199/QR639 DAC-DOH
- BA6355/QR655 CMB-DOH
- BA6366/QR654 DOH-CMB
- BA6374/QR904 DOH-MEL
- BA6375/QR905 MEL-DOH
- BA6387/QR956 DOH-CGK
- BA6395/QR955 CGK-DOH
- BA7003/QR3 DOH-LHR
- BA7004/QR4 LHR-DOH