In real life, Calamity Jane (Martha Jane Canary) never met 'Doc' Holliday: she actually was in love and pursued 'Wild Bill' Hickock. Doc Holliday did have a bad cough, as suggested in this episode, from his tuberculosis. He died in 1887.
In this episode Adam tells Doc Holliday that he's going somewhere "hot and dry" refering to Hell, but it's meant as a double entendre for Arizona where Doc went in hopes that the hot and dry weather would ease his tuberculosis.