Gul Dukat (Marc Alaimo) hated Star Trek: Deep Space Nine's Cardassian tailor, and former spy, Elim Garak (Andrew J Robinson) with a ion, and the feeling was mutual. Garak was exiled from Cardassia Prime and forced to live on Terok Nor. He remained on the station after it was ed into Federation hands and renamed Deep Space Nine, where he formed a close bond with Dr. Julian Bashir (Alexander Siddig) and continued to be a thorn in the side of Terok Nor's former prefect, Gul Dukat.
In Star Trek: Deep Space Nine's early days, Gul Dukat and Elim Garak had a strong dislike of each other, for reasons that weren't made immediately clear. It appeared to stem from the conflict between the Cardassian military and the secretive intelligence agency the Obsidian Order, of which Garak was a member. However, it wasn't until Star Trek: DS9 season 4 that the origin of Dukat and Garak's feud was revealed, and further complicated by the arrival of Dukat's daughter Tora Ziyal (Melanie Smith).
Why Gul Dukat & Garak Hated Each Other In Star Trek: Deep Space Nine
Dukat and Garak regularly feuded with each other in Star Trek: Deep Space Nine, from Garak aiding the investigation of a conspiracy involving Dukat in "Cardassians", one of Bashir's best DS9 episodes, to Garak ribbing Dukat about his inability to deactivate his own security systems in "Civil Defense". It's in that same episode that Garak also reminded Dukat about his failed attempt to assassinate the tailor while he was in charge of Terok Nor. The reasons for Dukat's attempts to kill Garak were rooted in the latter's work for the Obsidian Order and how it led to the death of Dukat's father.
As with many things about Garak, the truth of what happened to Gul Dukat's father is shrouded in secrecy. It's known that Gul Dukat's father was interrogated, tortured, and executed by the Cardassian government for an undisclosed crime. Dukat's father trusted Garak, something that Dukat felt was his only failing. Presumably, Garak used this trust to have the senior Dukat executed. The Obsidian Order were ruthless in their execution of those they deemed to be traitors to Cardassiq so Garak clearly won the trust of Gul Dukat's father to have this apparent traitor removed.
Garak's Relationship With Gul Dukat's Daughter Made Their DS9 Feud Worse
As Star Trek's executive producer Rick Berman vetoed Garak's Queerness, the Cardassian tailor was given a love interest in the form of Gul Dukat's half-Bajoran daughter, Tora Ziyal. It was seemingly one-sided, as Ziyal developed romantic feelings for Garak during the course of their close bond as Deep Space Nine's only Cardassian residents. This intensified the feud between the two Cardassian men, as Dukat threatened to kill Garak once again. In an attempt to poison his daughter's mind against Garak, Dukat revealed everything he knew about the death of his father, but Ziyal refused to give up on the man she had fallen in love with.
Furious at Ziyal's loyalty to the man who was responsible for her grandfather's death, Dukat disowned his daughter, a decision that would ultimately cost Tora her life, and Dukat his sanity. She remained aboard Deep Space Nine when it was retaken by the Cardassian-Dominion alliance in Star Trek: Deep Space Nine season 6. Ziyal's loyalties to the resistance led to her being executed by Legate Damar (Casey Biggs). The grief at Ziyal's death led to Garak and Dukat's paths diverging - Garak found redemption by becoming a hero of the Cardassian resistance, while Dukat doubled down on his anger and bitterness by becoming a violent religious zealot, ultimately leading to his undoing.