(covers information from several alternate timelines)
"Coup d'etat. It's French."
"Well, no stupid French thing will either."
The term is French for "a (sudden) blow (or strike) to a state" (literally, coup, hit, and État, state, always written with a capital É in this meaning).
In the mirror universe, Captain Jonathan Archer attempted to use the firepower of the USS Defiant to take over the Terran Empire in January 2155. He was ultimately deposed by his captain's woman, Lieutenant Hoshi Sato, who continued the plot and declared herself Empress. (ENT: "In a Mirror, Darkly, Part II")
In the 2250s, Gabriel Lorca attempted a coup against Emperor Georgiou. He failed, but managed to escape to the prime universe before returning to the ISS Charon in 2257 and finally being killed by Georgiou during his second attempt. (DIS: "Despite Yourself")
Minister Jaro Essa attempted a coup d'etat on Bajor in 2370. In the wake of the coup, led by Jaro's terrorist front, the Alliance for Global Unity - colloquially, the Circle - Jake Sisko and Nog were evacuated from Deep Space 9 on different runabouts. However, although he couldn't pronounce the word quite right, Nog assured Jake that nothing would keep the best friends apart. (DS9: "The Circle") The coup failed when it emerged that the Cardassian Union was covertly supporting the Circle with the intent of retaking Bajor, discrediting the movement, and the inhabitants of the station - including Jake and Nog - were reunited. (DS9: "The Siege")
In 2371, Kira Nerys told acting First Minister Kai Winn Adami that Shakaar Edon was not committing a coup by entering the next election for First Minister with the Bajoran Militia's support. (DS9: "Shakaar")
That same year, the Krajensky Changeling told Benjamin Sisko that the Tzenkethi autarch had been overthrown in a coup d'etat, although this was later proven to be a Dominion ruse to start another war between the Federation and the Tzenkethi. (DS9: "The Adversary")
In 2372, General Martok informed Captain Sisko and Lieutenant Commander Worf that the Klingon High Council supposed the recent coup on Cardassia, which had overthrown the Central Command, to have been engineered by the Dominion. Major Kira Nerys later mentioned that a coup on Cardassia was also possible without a Changeling influence since the demise of the Obsidian Order. (DS9: "The Way of the Warrior")
Also in 2372, Admiral Leyton attempted a coup d'état in the Federation starting on Earth, motivated by the Dominion threat. His plans involved reassigning loyal officers to key positions, simulating the passage of cloaked Dominion ships through the Bajoran wormhole, and arranging for Red Squad to sabotage Earth's global power grid. However, Captain Sisko and Odo foiled his plans. (DS9: "Homefront", "Paradise Lost")