Sort of.
We know something happened, but offhand
I don't recall any onscreen conflict:
The Borg Cooperative included former Romulans, Klingons, and Cardassians.
Additionally, in Unimatrix Zero we saw the Klingon Korok : The Ferengi are known to the Borg as Species 180 , so they have at least encountered the Ferengi.
I do not recall if any have actually been assimilated; they may not have been considered worthwhile material to add to the Collective.
As far as I know, no Dominion species have been shown as part of the Borg Collective.
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Yes, during the Borg Invasion of 2381 , there are non-canon invasions of the Borg attacking the entire Alpha Quadrant including the homeworlds of the Klingon Empire, the Romulan Star Empire and other planets.
This occurred in the novel: Star Trek-Destiny Book III:
Lost Souls
The Borg Invasion of 2381 was the historic attempt by the Borg Collective to completely exterminate the United Federation of Planets, Klingon Empire, and other allied worlds in the Alpha and Beta Quadrants and to subjugate the remaining worlds in local space.
It ended with the Collective being dismantled and subsequently absorbed into the Caeliar gestalt .
The invasion left dozens of worlds in the area of the Azure Nebula in utter ruin, decimating much of known space.
As a result of the invasion, upwards of 40% of the Federation Starfleet was destroyed, and over 63 billion citizens of the United Federation of Planets, Klingon Empire, Romulan Star Empire, Imperial Romulan State, and non-aligned worlds were killed.
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In addition to what other have said, the Borg attacked bases on both sides of the Romulan Neutral Zone.
The Memory-Alpha page for the TNG episode 1x26
"The Neutral Zone" says this:
This episode is the only one in Star Trek referencing the Borg attacking the Romulans.
In VOY: "Unity", Commander Chakotay of the starship USS Voyager encounteres liberated Borg drones in the Delta Quadrant nearly a decade later.
Among them was a former drone named Orum who identified himself as Romulan.
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The Borg attacked the outposts on the Federation-Romulan border during Season 1 of TNG.
The enemy which both Picard and the Romulan commander speak of during the season finale, which ripped their settlements from the surface, was confirmed as the Borg during the episode Q Who? .
So we know that some time after Annika Hansen's family was assimilated in the Delta quadrant, the Borg had at least basic knowledge of Alpha quadrant species and powers such as the Romulans, Klingons, Vulcans, Naussicans, etc.
They then set out towards the Alpha quadrant, and encountered the outposts of the Romulan Neutral zone, assimilating them.
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Here is a non-canon answer.
One of the main antagonists in the William Shatner novel
The Return is a assimilated Romulan named Vox .
He was depicted as a Romulan version of Locutus.
Vox was a 24th century Romulan individual who had been assimilated by the Borg to serve as a Speaker to the Romulan Empire.
Never thought the story made much sense for the Borg but it does exist.
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As far as I remember, when Borg cubes attack the alpha quadrant, they always engage the Federation and Earth is their primary target. Have there been any mention of any encounter between the Borg and the Romulans, Klingons, Cardassians, Ferengi or Dominion? Why target Earth when they could target Vulcan and weaken the federation?
Was there any mention in Star Trek canon or non-canon novels of any encounter between the Borg and Klingons or Romulans?
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<star-trek><star-trek-tng><borg><star-trek-eu>
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The Borg Invasion of 2381 was the historic attempt by the Borg Collective to completely exterminate the United Federation of Planets, Klingon Empire, and other allied worlds in the Alpha and Beta Quadrants and to subjugate the remaining worlds in local space. The Borg also attacked the outposts on the Federation-Romulan border during Season 1 of TNG. However, no Dominion species have been shown as part of the Borg Collective and it is not certain that any have actually been assimilated. They may not have been considered worthwhile material to add to the Collective. The Borg invasion of 2381 is significant in that the Klingon Empire, the Romulan Star Empire and other planets were attacked.
The Borg invasion of 2381 is significant in that the Klingon Empire, the Romulan Star Empire and other planets were attacked.