Jett Reno declares to Dr. Hugh Culber "I'm an engineer not a poet", which is an homage to Dr McCoy's "I'm a Doctor not a ______ (fill in the blank)"
When Reno enters the sickbay she declares a medical emergency which turns out to be a hangnail. In The Menagerie: Part I (1966) Bones is summoned aboard the Enterprise under the pretext of a medical emergency and claims that 'Someone probably just found a hangnail'.
Mia Kirshner (Amanda) plays the mother of Ethan Peck (Spock). In real life, Kirshner is only 11 years older than Peck.