This question is really about matching each organism to its arthropod vector rather than just its name.
Rickettsial pox: caused by Rickettsia akari, spread by mites living on house mice, not a tick.
Epidemic typhus: caused by Rickettsia prowazekii and spread by the human body louse.
Q fever: caused by Coxiella burnetii, which circulates among cattle, sheep, and goats through a tick-borne cycle in nature, making it the tick-transmitted disease among the four.
Trench fever: caused by Bartonella quintana and, like epidemic typhus, spread by the body louse.