(covers information from several alternate timelines)
A delusion was a psychological disorder that displayed a false belief that was not necessarily visible to the believer (unlike a hallucination), often brought on by mental illness.
In 2258 of the alternate reality, Doctor Leonard McCoy referred to James T. Kirk as being delusional due to a Melvaran mud flea vaccine and took responsibility for his condition. (Star Trek)
In 2268, Korax accused James T. Kirk of having "delusions of godhood." (TOS: "The Trouble with Tribbles")
In 2269, while Kirk was trapped in Janice Lester's body, Doctor Arthur Coleman diagnosed Lester with the delusion of being Captain Kirk. (TOS: "Turnabout Intruder")
In the script of TOS: "Court Martial", Samuel T. Cogley referred to Benjamin Finney as having suffered "delusions of persecution" which had led Finney to try to frame Captain Kirk.
In 2371 of the mirror universe, Jadzia Dax described Terran rebels as "ex-slaves with delusions of grandeur", expressing the futility of continuing to fight against the Terran Empire. (DS9: "Through the Looking Glass")