Your question asks if Florida became a state during the time of George Washington's first term.
In this case, Your answer would be FALSE
Florida became a state in the United States way after George Washington finished his first term. Florida became a U.S state during the year of 1845, while Washington's first term ended in 1792. This concludes your answer that Florida didn't become a state when George Washington was in office. Florida would've just been a territory. Florida would be "owned" by Spain during the time of George Washington's term.