The short answer is no. The idea of the quick draw duel is fiction that was written about in "dime novels". There are a couple incidents through western history that come close to a showdown that these writers based their stories on.

One of the first incidents to take place involved Davis Tutt and James Butler Hickok, better known as Wild Bill. It's 1865 and Hickok finds himself in Springfield, MO after the war. He was good friends with a former Confederate named Davis Tutt, until they have a falling out over cards and women. Tutt had loaned Hickok money to cover gambling debts, and, while playing cards with Hickok, demanded repayment for the loan. Hickok, actually having won money, pays the debt back immediately. Tutt then tells Hickok he owes money on another debt, but the amount is disputed. Tutt not liking this takes Hickok's pocket watch off the table and tells Hickok he is taking it for payment. Hickok tells him that if he sees him wearing it will be a dead man walking around with it. The next Tutt is crossing the square in Springfield showing off Hickok's fancy watch. Hickok tells him to remove the watch or he's going to die. Tutt draws his pistol and fires missing Hickok. Hickok draws and returns fire striking Tutt in the chest, right through his heart, killing him. The shot was made at a distance of 75 yards, and there is still a plaque noting the incident. Harper's Weekly got a hold of the story and ran with, turning Hickok in to a household name. Soon there were stories of duels at high noon appearing all over.

So yes gunfights happened, but they were spontaneous events, not planned out like Hollywood likes to show.