The Colt Walker revolver was a cap and ball revolver, meaning loose black powder was poured in the the guns cylinders; then a round lead ball was placed on top of the powder and loaded with the revolver's loading ram. A percussion cap ignited the black powder when the hammer dropped on it. There wouldn't be a Colt Walker "cartridge" as this gun never used cartridges.
The Colt "Walker" was produced in 1847 not 1838, was a cap and ball (percussion cap, loose powder and lead ball) pistol, and did not have a cartridge made for it. Further, the weapon is said to have been used in the Texas War of Independence, which was over by 1836. In 1838, the only Colt revolver available would have been the Paterson, quite a few of which were used by Texas Rangers of that period.
Near the beginning of the episode Paladin and Bartlett are looking at the bullets and Paladin said one of the bullets was a Walker 1838. Bartlett said that gun was used in the Texas war for independence. The Texas War lasted from 1835 to 1836, two years before the pistol was used.