A creature similar to a Maravel dragon
A two-headed mechanical dragon
Da Vinci's dragon with a Vulcan ear
A dragon was a type of winged serpent-like reptilian animal. Such animals were known to live on Berengaria VII and Maravel. As the result of extra-terrestrial contact, Earth mythology also featured dragons.
A Vulcan survey team to Berengaria VII made note of a species as one of flying reptiles, over two hundred meters in length, that breathed fire. A century later, Spock had seen a dragon when he visited the planet. (ENT: "Bound"; TOS: "This Side of Paradise")
In 2269, James T. Kirk saw a species of flying dragons in the Lactra VII Zoo, in the rainforest environment, similar to Maravel dragons. (TAS: "The Eye of the Beholder")
Also in 2269, Hikaru Sulu and Leonard McCoy were chased by a two-headed, fire-breathing mechanical dragon on the Shore Leave Planet when the planet's master computer was attempting to "turn off" the Enterprise landing party. After the master computer agreed to cease its pursuit, McCoy and Sulu were seen on the master computer's viewscreen, enjoying a picnic with the dragon, Alice, and the White Rabbit. (TAS: "Once Upon a Planet")
The Centerplace on Ligon II featured a sculpture of an animal similar to a Chinese dragon. (TNG: "Code of Honor")
In Earth culture
On Earth, dragons were more commonly known as supernatural monsters or deities in the myths of the Native Americans, Europeans and Chinese. In 2270, it was discovered that the basis of these myths was Kukulkan, an immensely old alien being who had visited all the nations on Earth in the distant past. (TAS: "How Sharper Than a Serpent's Tooth")
A species of large reptiles on Earth were called Komodo dragons, though they did not have wings. (TNG: "New Ground")
An ancient Greek myth told of a dragon that fell in battle, whereupon its teeth took root in the ground and sprang back as warriors to continue the fight. In 2376, Chakotay compared the hundreds of bio-pods containing the last survivors of the Vaadwaur race to dragon's teeth. (VOY: "Dragon's Teeth")
In Medieval times, dragons, sea serpents and other mythological creatures marked unexplored regions on maps. If a ship were to sail too far out into the ocean, it was said that crews might threaten to hang their captain from the yard arm if he refused to turn back. (TNG: "Where Silence Has Lease")
In 1999, Jason Janeway owned a t-shirt with a red dragon on it. (VOY: "11:59")
The holographic Leonardo da Vinci was so fascinated by Tuvok's Vulcan ears that he drew a picture of a dragon with a pointed Vulcan ear as a tail. (VOY: "Concerning Flight")
The Doctor compared Qatai's quest to kill the "telepathic pitcher plant" to dragon-slaying. (VOY: "Bliss")