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Hello. I'm thinking of travel to Doha from Miami on June 6th, flight QR778.
These are the codes:
J4 C4 D4 I4 R0 P0 U4 Y4 B4 H4 K0 M0 L0 V0 S0 N0 Q0 T0 G4 O0 W0 E4 X0
Is there any way of knowing if there are business class seats open?
Thanks!

J C D I R and P are the business cash fare codes, U is the business award fare code.

That flight is either close to
sold out, or there is an error. U fares might not be expected given everything else. I'd check again a bit later,. Alternatively search for revenue fares at QR website and see what comes up.

On Expertflyer QR778 6 june:
J9 C9 D9 I9 R9 P9 Y9 B9 H9 K9 M9 L9 V9 S9 N9 Q0 T0 O0 W0

That looks very different from your source

I have been searching award space from the US to DOH and it seems like MIA often has availability, so I would keep checking. If you are using AA miles, it does seem that many times they require you to have a married segment. For example, I will sometimes see DCA-MIA-DOH available, but sometimes not MIA-DOH. Not sure why since the first segment is AA.