He was born in Montreal on 13 July 1887, the son of banker James Baxter and his wife, Hélène de Lanaudière Chaput.
Quigg was educated by Jesuits at Loyola College,a private boys school in Montreal. He joined the Montreal Amateur Athletic Association when he was 17 and quickly earned a reputation as a star football and hockey player. He played hockey with the Montreal Shamrocks until he took a low blow to his eye from a stick during a game in 1907, and lost the sight of it. Although he was no longer able to play hockey, he coached, and is credited with organizing one of the first international hockey tournaments ever played in Paris.
In 1911 he dropped out of his first year in Applied Sciences at McGill University to accompany his mother and sister (Hélène Douglas) to Europe. While in Brussels, he met and fell in love with a 24-year old cabaret singer, Berthe Mayné. He was determined to bring Mayné back to Montreal with him, and booked her into a stateroom of her own on Titanic C-90 under an assumed name, Mme. DeVilliers. Baxter himself occupied boarded the Titanic at Cherbourg with Ticket No. PC 17558.
On the night of the sinking, he was in his cabin (B-58) when his mother demanded to know why the Titanic had stopped in mid ocean. When he stepped outside just before midnight to investigate, he saw Captain Smith talking to Bruce Ismay outside Ismay's cabin next door. "There's been an accident, Baxter, but it is all right," Smith told him. As Smith hurried away to the bridge, Ismay told him to get his mother and sister into the lifeboats. Baxter carried his mother up the grand staircase to lifeboat 6. "Quigg didn't seem at all disturbed," his sister later told The Montreal Standard, "While he didn't relish being parted from us, he bade me farewell bravely." As he put his mother into the boat he handed her a sterling silver flask of brandy, and she began to complain about his drinking. He cut her short: "Etes vous bien maman?" he asked, Au revoir, bon espoir vous-autres." (Goodbye and keep your spirits up everyone.) Berthe Mayné didn't want to get into the boat without him, but Molly Brown convinced her to do so.
He waved them away. One of the last survivors to get off the ship saw Quigg heading towards his room and then having water come into the hallway, the passenger looked, and the room had flooded from the open window. The survivor left, and Quigg drowned in the sinking, his body, if recovered, was never identified.
Quigg Baxter is remembered on his mother's grave at Notre Dame de Neiges Cemetery, Montreal.