A red carpet was a symbol used to welcome government officials or people in high power for formal diplomatic relations.
In 2152, Captain Jonathan Archer claimed to Commander Charles Tucker III that Zobral was preparing to "roll out the red carpet" for them when he invited the two men to visit his homeworld. (ENT: "Desert Crossing")
In 2364, the Ligonian welcoming party beamed aboard the USS Enterprise-D in one of the ship's cargo bays. One of the Ligonians rolled out a red carpet on the floor of the cargo bay shortly before Lutan, the leader of the Ligonians, was brought aboard. (TNG: "Code of Honor")
In 2369, shortly before the Wadi, a species from the Gamma Quadrant, made First Contact with the United Federation of Planets on Deep Space 9, Commander Benjamin Sisko dressed in a Starfleet dress uniform for the first time in three years. His son Jake commented on his father's wearing of the dress uniform and the impending first contact as akin to "rolling out the red carpet, huh?" (DS9: "Move Along Home")
In 2373, Jadzia Dax told Kira Nerys while welcoming Tekeny Ghemor to Deep Space 9 that there must be a red carpet lying around somewhere, referring to Kira's elaborate welcome to him when he visited the station shortly before his death. (DS9: "Ties of Blood and Water")
In 2374, Legate Damar and Weyoun attempted to make peace between the Federation and the Dominion to end the Dominion War. Captain Benjamin Sisko was to officiate the peace offering on Deep Space 9. In Ops, he announced to his senior staff to "roll out the red carpet." However, the peace offering was later found to be a ruse for the Dominion to gain control of the Kabrel system and procure more Ketracel-white for the Jem'Hadar. (DS9: "Statistical Probabilities")