The metric system was a measuring system used by the replicators aboard Federation starships during the mid-24th century. (TNG: "The Vengeance Factor", "The Defector", "Future Imperfect", "Ensign Ro", "Realm Of Fear"; DS9: "Call to Arms")
On Earth, the metric system was one of the primary systems used, it was also used by other alien species including the Akaali, the Brekkians, the Ferengi, the Retellians, Tessic's species, the Xindi, and the Zibalians, among others. (Star Trek: The Next Generation, Star Trek: Deep Space Nine, Star Trek: Enterprise)
When the Romulan Admiral Alidar Jarok, who defected to the USS Enterprise-D in 2366, he requested from the ship's replicator a glass of water at the temperature of twelve onkians. The Enterprise computer was not familiar with that unit, and informed Jarok that "[t]his system is calibrated to the Celsius metric system." The clearly irritated Jarok then clarified, "[a]ny temperature at all on the cold side of whatever your system is." (TNG: "The Defector")
It was never made clear if the usage of the metric system on screen by alien species indicated that it was a universal concept, or if it was an example of implicit translation or general equivalency, achieved through the universal translator. The reference from "The Defector" would be one of a small number of instances that might suggest otherwise.
The needs to make a necessary conversions was implied by statements such as the one where Kirk remarks to Abraham Lincoln that the Enterprise could convert to minutes in TOS: "The Savage Curtain", and who also spoke the Lincoln figure using Imperial measurements.