So here’s how I imagine the scenario going down. Let’s say the Death Star and a Borg Cube meet in high orbit over Abydos. Both vessels are alone and unaided by their respective nations, carrying their full typical complement of personnel, weapons, ammunition, technology etc. Both are informed “There is a large enemy ship approaching” and given no other details.

THE BORG CUBE

A Borg cube dwarfs a Galaxy-class starship, perhaps a couple kilometers to a side. Borg Cubes are equipped with fairly typical Star Trek phaser arrays that punch about at their weight class, a single Borg cube is more massive than and more than a match for a Federation battle fleet. The Cube contains many thousands of drones that are individually capable of assimilating non-Borg people, which is typically their main goal. Borg cubes tend not to be equipped with weapons of mass destruction, nor do they carry anything resembling fighters or interceptors. Borg cubes are equipped with Trans-warp drive and are able to travel many thousands of times faster than the speed of light. They are also equipped with transporter technology which is entirely unfamiliar in the Star Wars universe. Deflector shields in the Star Trek universe seem to be more capable of outright stopping damage, though they can be overwhelmed, causing them to fail. The Borg are able to quickly measure the nature of any attacking weapon and adapt their shields to be impervious to that weapon after a few attacks.

THE DEATH STAR

A space station of truly astronomical size, the Death Star is mistakable for a dwarf planet or moon, though it is much less massive being mostly hollow. The Death Star is staffed by squadrons of TIE fighters and an army of storm troopers equipped with blaster rifles. The Death Star’s surface is studded with point defense lasers, and much of the upper hemisphere is occupied by the crater-like dish of the main turbolaser, which is capable of blasting an Earth-like planet to boulder-sized chunks in a matter of seconds. The Death Star is a weapon of mass destruction/ultimate terror. Personal teleportation is unknown in the Star Wars universe and the Death Star is not designed to defend against them. Deflector shields exist in the Star Wars universe though they seem less important or effective. The Death Star is equipped with hyperdrive allowing it to travel at some speed faster than light; the original Death Star flew from the Alderaan system to the Yavin system under its own power. Both Death Stars are depicted as fairly easily destroyed by weapons carried by fairly small craft, damaging the central power system will cause it to explode with enough force to blast the entire space station to smithereens.

THE BATTLE

The Borg could win this fight by warping in very close with their vastly superior speed and immediately transporting many hundreds of drones aboard to assimilate the confused and unprepared crew, especially if targeting the upper command staff. The storm troopers may have early success fending off the invaders, until their weapons suddenly stop being effective.

They could, but probably wouldn’t. The Borg tend to prefer to hang back and taunt their opponents over the radio for a minute “We are the Borg. Resistance is futile. We will add your uniqueness and distinction to our own.” etc. Their whole opening monologue is enough time to prepare and fire the Death Star’s turbolaser, which would effortlessly obliterate the Cube and the moon behind it.

The final winning move here would be to jump to hyperdrive as soon as possible to avoid any issues with individual drones that may have survived who might make it to the Death star and start a First Contact “they’ve assimilated deck ten” scenario, which I honestly don’t see the Empire winning without destroying the Death Star. The Empire isn’t as good at treknobabble as the Federation, so Grand Moff Tarkin isn’t going to confuse the Borg to death with a paradoxical equation or whatever.

If a Sith is present, he will probably hold his own with a light saber until the Borg adapt, and then make his way to a waiting escape ship using Force powers to throw drones out of the way.

POST SCENARIO

Should the Death Star make a clean escape, a follow-up encounter with another Borg cube would likely not go as well. They wouldn’t have had a chance to witness the Borg’s adaptation ability or indeed learn much about them. They probably destroyed the first cube with a less than full power shot, and trying that again might not work this time. It is my understanding that the turbolaser requires time to recharge between shots, and this may be enough time for the second cube to move in and begin their attack/assimilation. The Death Star is many orders of magnitude slower than a Borg cube and unable to escape.

The Death Star has no hope against two Borg cubes; it may destroy one while the other successfully closes to deploy drones.

A Borg cube is much smaller than the Death Star, and seeing the Death Star survived a collision with a super star destroyer, I imagine a direct collision with a Borg Cube would only admit entry by surviving drones.

A Goa’uld pyramid ship is by far outclassed by both vessels. SG-1 would take down the Borg cube with three men, one woman, four P90s and a kilogram of naquadria, though Daniel Jackson would die at least twice. There’d be a really uncomfortable body horror scene where a goa’uld symbiote and Borg nanotechnology fight for control of a Jaffa.

And here comes Carter with the folding chair exploding sun!

You know it’s all the same people in all three groups, right? (/s)

Replicators would take out both, even if Borg Cube and Death Star joined forces.

Everyone’s all like “but the Death Star destroys planets”. Yeah, but they’d have to hit the Borg cube - significantly smaller and way more maneuverable. In the end, it comes down to how easy Empire tech is to assimilate. If it comes to a fire fight, we all know storm troopers can’t target for shit.