My question is the exact same as the title.
Not really in the traditional sense of the word "refugee." The United States is one country and acts as one. The states cooperate with each other and aren't really "states" in the international (international meaning not U.S. here) sense of the word.
Particularly relevant to this conversation is the "Extradition Clause" of the U.S. Constitution:
A person charged in any state with treason, felony, or other crime, who shall flee from justice, and be found in another state, shall on demand of the executive authority of the state from which he fled, be delivered up, to be removed to the state having jurisdiction of the crime.
In other words, you can't, as an example, commit a crime in one state and seek refuge in another state. The authorities will deliver you back to the state you came from. So you can't be a refugee in that state in that sense.