Parson Weems (1759-1825) wrote in The Life of Washington (1800) the tale that a young George Washington told his father, “I cannot tell a lie, I did it (chopped down a cherry tree) with my little hatchet.” The humorous Texas version has it that George Washington lived in Texas and chopped down a cactus. When Washington admitted it and told his father that he could not tell a lie, the family moved to Virginia because all true Texans can lie/brag better than that.
 
The Texas version of the tale dates in print to 1955.
 
   
Wikipedia: Parson Weems
Mason Locke Weems (October 11, 1759 – May 23, 1825), generally known as Parson Weems, was an American book agent and author. He is best known as the source of some of the apocryphal stories about George Washington. The famous tale of the cherry tree (“I cannot tell a lie, I did it with my little hatchet”) is included in The Life of Washington (1800), Weems’ most famous work. This nineteenth-century bestseller depicted Washington’s virtues and provided an entertaining and morally instructive tale for the youth of the young nation.
   
Google News Archive
27 July 1955, Milwaukee (WI) Sentinel, Earl Wilson column, pt. 1, pg. 15, col. 4:
TEXAS NOW says George Washington was a native son. As Tommy Griffin of New Orleans relates it, little George cut down a big cactus plant. “George,” said his father, “did you cut down this cactus plant?” Little George whimpered, “I cannot tell a lie. I did it with my little hatchet.”
 
His father roared: “If you can’t lie any better’n that, get out of Texas!”
 
So George sneaked over to Virginia and grew up there, because Texas couldn’t tolerate such untalented prevaricating.
 
Google Books
November 1955, Boys’ Life, “Think and Grin,” pg. 98, col. 1:
Texas now claims George Washington as a native son. According to Texans, he once cut down a big cactus plant.
 
“George,” asked his father, “did you cut down this cactus plant?”
 
Little George whimpered, “I cannot tell a lie. I did it with my little hatchet.”
 
His father roared: “If you can’t tell a better story than that, son, get out of Texas!”—Barry Patton, Salinas, Calif.
 
Google Books
Myth and the History of the Hispanic Southwest
By David J. Weber
Albuquerque, NM: University of New Mexico Press
1988
Pg. 139:
There is a cartoon that represents a Texas variation of the story of George Washington and the cherry tree. It shows young George Washington and his father in Texas.
 
“I cannot tell a lie father,” George says, “I chopped down the cactus.”
“Well George, we’re going back to Ol’ Virginia,” his father replies. “If you can’t tell a lie, you’ll never make a true Texan.”
     
Google Books
Country Music Humorists and Comedians
By Loyal Jones
Urbana, IL: University of Illinois Press
2008
Pg. 80:
Here is one of Red’s (Red Blanchard, 1919-1980—ed.) routines, remembered by David Wylie:
 
George Washington was not born in Virginia. He was born in Texas, and he chopped down a cactus, and his father said “George, did you chop down that cactus?” He said, “Yes, Dad, I cannot tell a lie.” His father said, “If you cannot tell a lie, we’d better get out of Texas.” They moved to Mount Vernon.
 
The Notes:
Ronald Reagan’s Private Collection of Stories and Wisdom
By Ronald Reagan
Edited by Douglas Brinkley
New York, NY: HarperCollins
2011
Pg. 248:
Texas even claims Geo. W. was a Texan. Story is he cut down a cactus—his father said Geo. did you cut that cactus down. Geo. whimpered I cannot tell a lie—I did it with my little hatchet. His father yelled—if you can’t tell a better story—get out.