A Dodge is a make of car, so it would mean to get out of the car, but I wonder if perhaps the phrase was "Get out of Dodge". Is this something you read, or something you heard? If it was actually "Get out of Dodge", then it means to leave a place and it refers to Dodge City, Kansas and comes from old cowboy movies.

From urbandictionary.com: To leave somewhere immediately, to evacuate or scram. "Get the hell out of Dodge" is a reference to Dodge City, Kansas, which was a favorite location for westerns in the early to mid 20th century. Most memorably, the phrase was made famous by the TV show "Gunsmoke," in which villians were often commanded to "get the hell out of Dodge." The phrase took on its current meaning in the 1960s and 70s when teenagers began to use it in its current form. ----- For your sentence, Dodge is a car brand, so "the Dodge" could mean to get out of the car.