Captain James T. Kirk (William Shatner) has not been referenced by Star Trek: Discovery but Captain Michael Burnham (Sonequa Martin-Green) is essentially her show's version of Kirk. About to enter season 5, Star Trek: Discovery shed its mid-23rd-century prequel setting and set up shop in the distant 32nd century starting with season 3. It was a crucial move to remove Star Trek: Discovery from its original era about 10 years before Kirk becomes Captain of the USS Enterprise in order for Burnham to forge new canon in the far future.
Captain Christopher Pike (Anson Mount) and Spock (Ethan Peck) alleviated some complaints, but there was no mention of James T. Kirk.
Why Kirk Isn't Mentioned In Star Trek: Discovery
Star Trek: Discovery's timeline in season 1 and 2, and its subsequent departure for the 32nd century in season 3, neatly avoided mentioning Captain Kirk. Discovery seasons 1 and 2 were set from 2256-2258, when Kirk was a Lieutenant and years away from becoming Captain of the Enterprise. Further, neither Captain Pike nor Spock had met Jim Kirk yet before the USS Discovery time traveled permanently to the far future. Naturally, Star Trek: The Next Generation.
However, there remains no logical storyline reason as yet for Star Trek: Discovery to reference Captain Kirk. Star Trek: Discovery's 32nd century is 900 years after Kirk's heyday of saving the galaxy as Captain of the Enterprise, but by Burnham's time, he was just one of many heroic Starfleet Captains. It's possible Michael looked up Spock's Starfleet career after she left and learned of his adventures with Kirk, but that hasn't been canonically confirmed on the show. Star Trek: Strange New Worlds.
Michael Burnham Is Discovery's Version Of Kirk
Star Trek: Discovery doesn't need to reference Kirk because they have Michael Burnham, who exhibits many of the legendary Captain's traits. Burnham's core heroism, her risk-taking, her disregard of regulations when they oppose her morals and clarity of purpose, and her propensity for leaping where angels fear to tread are all derived from Kirk's archetype. Granted, Burnham seems to borrow more from Chris Pine's Abrams movie Kirk than William Shatner's, but Michael and Jim are both fiercely intelligent, and they also share a deep comion, a strong sense of duty, and loyalty to their crew.
Like Kirk, Captain Burnham is Starfleet's go-to whenever the Federation is in crisis or a threat to the galaxy emerges. Michael also makes grave mistakes and was court-martialed, as Kirk has. Of course, Kirk and Spock were like brothers while Spock was Michael's adoptive brother. Now that Kirk has a role in Star Trek's Paramount+ series, it's possible he will be mentioned in Star Trek: Discovery, but Captain Burnham will continue to embody many of the familiar aspects that made Kirk a legend.
Star Trek: Discovery Season 5 is expected to premiere in 2023 on Paramount+.