|An artist’s illustration of a spacecraft’s escape trajectory (bright white line) from our solar system into interstellar space. Credit: Mike Yukovlev Johns Hopkins Applied Physics Laboratory - Link 2 below|
Topics: Astrophysics, Interstellar Travel, NASA, Spaceflight, Star Trek
Yes, an actual slingshot effect does exist.
As much a fan as I am of the Trek, this isn't it.
When a spacecraft in orbit about a primary body comes close to a moon that is orbiting the same primary body, there is an exchange of orbital energy and angular momentum between the spacecraft and the moon. The total orbital energy remains constant, so if the spacecraft gains orbital energy then the moon's orbital energy decreases. Orbital period, which is the time required to complete one orbit about the primary body, is proportional to orbital energy. Therefore, as the spacecraft's orbital period increases (the slingshot effect), the moon's orbital period decreases.
But because the spacecraft is much, much smaller than the moon, the effect on the spacecraft's orbit is much greater than on the moon's orbit. For example, the Cassini spacecraft weighs about 3,000 kilograms, whereas Titan, the largest of Saturn's moons, weighs about 1023 kilograms. The effect on Cassini is thus about 20 orders of magnitude greater than the effect on Titan is. 
It would begin in the early 2030s, with a launch of a roughly half-ton nuclear-powered spacecraft on the world’s largest rocket, designed to go farther and faster than any human-made object has ever gone before. The probe would pass by Jupiter and perhaps later dive perilously close to the sun, in both cases to siphon a fraction of each object’s momentum, picking up speed to supercharge its escape. Then, with the sun and the major planets rapidly receding behind it, the craft would emerge from the haze of primordial dust that surrounds our star system, allowing it an unfiltered glimpse of the feeble all-sky glow from countless far-off galaxies. Forging ahead, it could fly by one or more of the icy, unexplored worlds now known to exist past Pluto. And gazing back, it could seek out the pale blue dot of Earth, looking for hints of our planet’s life that could be seen from nearby stars.
All this would be but a prelude, however, to what McNutt and other mission planners pitch as the probe’s core scientific purpose. About a decade after launch, it would pierce the heliosphere—a cocoonlike region around our solar system created by “winds” of particles flowing from our sun—to reach and study the cosmic rays and clouds of plasma that make up the “interstellar medium” that fills the dark spaces between the stars. Continuing its cruise, by the 2080s it could conceivably have traveled as far as 1,000 astronomical units (AU), or Earth-sun distances, from the solar system, achieving its primary objective at last: an unprecedented bird’s-eye view of the heliosphere that could revolutionize our understanding of our place in the cosmos. 
1. How does the slingshot effect (or gravity assist) work to change the orbit of a spacecraft? Scientific American, July 11, 2005
Jeremy B. Jones, Cassini Navigation Team Chief at NASA's Jet Propulsion Laboratory
2. Proposed Interstellar Mission Reaches for the Stars, One Generation at a Time
Scientific American, Lee Billings, November 2019