Sure, there can be Klingon engineers and designers, but the Hur'q had them under their thumbs for a long time, so at this point I'd say the ancient designs we saw in DSC would be the best candidates for Hur'q designs.
In terms of onscreen Trek, the Hur'q never had anybody under their putative thumbs, least of all Klingons. They just stole the Sword of Kahless and fled.
Might of course be they did enslave the Klingons for millennia first, and the Klingons just don't discuss it with outsiders.
Yeah, but we know both the Klingons and Romulans operated their own respective birds of prey in the 22nd century which were similar to their ships of the next century, so I'd say that the name is just coincidence. Perhaps it's just the different words for "small covert raider" in both Klingonese and Rihannsu happen to be rendered in English best as "Bird of Prey"?
Again in terms of onscreen Trek, the Romulans never had ships named Bird of Prey. In "United", they had an unseen design called Warbird, the propulsion system of which was used as the basis for the drone that masqueraded as assorted adversary vessels, but that's pretty much it.
Warbird was the actual ship designation shared by Romulans and Klingons. The former had feather patterns to their distinctly un-birdlike ships, while the latter flew ships that had wings, necks and heads. The Orions did that, too; it seems pretty natural to make flying things in the image of flying beasts, I guess.
The menagerie of Klingon ships we saw in S1 might have featured its share of non-warships: many of the more esoteric designs only appear in the Battle of Binaries, which wasn't a preplanned engagement but started as an impromptu meeting of closest ships from a number of important Houses. Not necessarily all 24 of them, mind you, as only half a dozen Klingons actually contact T'Kumva; perhaps there were only as many Houses as there were distinct designs, perhaps even fewer.
The likely warships would be the scythe-winged, skull-headed Birds of Prey, which appear in virtually every engagement (perhaps as mere auxiliaries to the big ships), and the distinctive
Qugh destroyers which are the first to respond at the Binaries and pose the biggest threat to Lorca, plus the two-nacelled, winged ships that look like melted
K'Tingas and are seen threatening Earth. The rest might be "civilian" ships (for the Klingon value of the word), perhaps even generic alien ones (explaining how Mudd's ride is based on one).
Timo Saloniemi