...While I know it will live forever in the hearts of much of the Quest, there is nothing In the world I believe in less than the idea that the Klingons care at all about our emotional support dreadnought.
It was a ship from a century ago from, when Earth was a distinctly peripheral power. Furthermore, in terms of is construction the Thunderchild reflects a defensively-focussed design philosophy which the Klingons would view with contempt. The Skate or the Selachii they would probably rate slightly more highly as unambiguously offensive weapons, but still view as somewhat compromised by Federation weakness. But also realistically only the Selachii is going to slightly figure in Klingon imagination as "that old patrol ship we see sometimes".
Probably their equivalent of naval history nerds or hardcore wargamers might recognise the name, but it's fundamentally a piece of historical trivia from a time when Earth was just not on their radar at all. A Vulcan ship of the same time period might elicit slightly more reaction, but only slightly. It would be like the modern People's Army Liberation Navy having a visceral emotional reaction to the the memory of the Royal Hellenic Navy armoured cruiser Georgios Averof.