A potential Starfleet recruit asks if transporter crewmen ever get to sit on the bridge, which Mariner says could happen. The most famous transporter chief in Star Trek, Miles O'Brien, was first introduced as an unnamed crewman sitting on the bridge of the Enterprise-D.
The archaeologist tells a potential recruit that they're signing up for "7 years in a windowless room." In the 80s-00s, most Star Trek shows ran for 7 years.
Before fading away, the younger Rutherford places his hand to the older Rutherford's temple and tells him to, "Remember." This echoes Spock's actions at the climax of Star Trek II.
The "truthers" suggest that Starfleet admirals are being controlled by "alien butt-bugs," which Boimler dismisses as just a "conspiracy." Mariner informs him that had actually happened, a reference to the penultimate episode of the first season of Star Trek The Next Generation, titled "Conspiracy." However, the alien parasites in that episode burrowed themselves into the base of the hosts' skulls, they did not crawl in through their butts.
At the Starfleet recruiting booth, Boimler and Mariner have a cutout of Kirk and Spock from the 1970s Star Trek Animated Series, as well as a model of a Constellation-class starship (similar to the one Picard kept in his ready room, though Picard's was gold rather than the gray seen here).