There are a lot of similarities between DC's Green Lantern Corps and Star Trek's Starfleet. Both operate in space, are color-coordinated and are prominently feature larger-than-life militaristic organizations. While the two groups have been around for awhile, it wasn't until the two actually crossed over, appearing alongside each other in Mike Johnson and Angel Hernandez's Star Trek/Green Lantern.
The first part of Star Trek/Green Lantern, which is titled "The Spectrum War," essentially deals with an alternate version of 2009's Blackest Night event in which Nekron wiped out the DC Universe. However, before Nekron could obtain total victory, Ganthet opened up a portal to a version of the Star Trek universe's Kelvin Timeline and sent six power rings through, which also had the effect of pulling through a number of surviving lanterns. Various Star Trek characters get their own power rings, and the remaining Lanterns eventually team with the crew of the Enterprise to take down Nekron on the dead planet Vulcan. During their battle with Nekron, Spock becomes a White Lantern by donning all of his compatriots' rings, and the team is finally able to take the villain down once and for all by accessing the Life Entity.
The next part of the crossover was Stranger Worlds. In that event, the various DC characters have integrated more into the Star Trek universe. However, a number of surviving DC villains remain to inflict terror on others, including Sinestro. Eventually, a newly escaped Khan obtains a Red Lantern ring, which leads to another showdown with the crew of the Enterprise and their allies. During the battle, Khan nearly kills Captain Kirk. However, the hero received a Green Lantern ring when things seem darkest, allowing him to defeat his adversary. As the crossover concludes, the remaining Lanterns and Enterprise crew head off to look for other power sources and planets, starting with Superman's homeworld of Krypton.
Much of Johnson and Hernandez's work explores the effects of Lantern technology in the Star Trek world, and several Enterprise heroes end up getting rings of their own. Chekov, Uhura and McCoy are chosen by the Blue, Violet and Indigo rings, respectively. Over the course of the story the three prove themselves capable of the great hope, love and compassion their respective rings symbolize and draw from. This is also particularly surprising in the case of McCoy, as he's not exactly known for his warmth. Outside of the main crew, various other characters get Yellow, Red and Orange rings. However, for the most part, the Star Trek characters -- with the exception of Khan -- don't pose nearly as much threat as DC villains Sinestro, Atrocitus and Larfleeze.
However, a big part of any crossover is getting to see characters from different universes interact, and Star Trek/Green Lantern provides a lot of novel interactions. There are moments where Captain Kirk and Hal Jordan absolutely appear to be kindred spirits, particularly when Spock makes an observation about how both act without thinking. Overall, the partnership between the Lanterns and the Enterprise inevitably benefits both parties in a lot of interesting ways, proving the two franchises work very, very well together -- even if there don't seem to be any plans for another crossover in the future.