Parmen and the other natives of Sahndara called themselves Platonians and their new planet Platonius because they claimed to follow Plato's philosophy, but Spock noted that Plato wanted justice, which the Platonians did not seem to care too much about. (TOS: "Plato's Stepchildren")
In sharing philosophical teachings with the Drayans, Kathryn Janeway pointed out Plato, who she described as one of mankind's greatest philosophers, who wrote "that what we see around us are only poor shadows of ideal objects which exist on a higher plane." The Drayans held similar beliefs, as shared by Alcia, were that they "believe that physical matter is only an illusion. The body is not the true self, only a representation." She further revealed that their connection to a "'higher plane' is more important than [their] attachment to this brief existence, however real it may seem." (VOY: "Innocence")
Vulcan philosophers were known to study the works of Plato, as evidenced in the work A Cave Beyond Logic: Vulcan Perspectives on Platonic Thought. (VOY: "In the Flesh")
In Diane Duane's novel Dark Mirror Plato's mirror universe counterpart is mentioned as writing that the perfect government is one that metes out fear in proper proportion to the people. In the mirror universe such a work would likely be titled "Kingdom and Empire" rather than "The Republic" as in the real universe.