Simpson took this story from Aesop's Fables

That's what I would have bet too. The story sounded so familiar to me. But it is my nature to check things. A site claiming to have all of Aesop's fables had several fables with turtles and several with asps but none with both. I got curious and I found no mention of any story about a turtle and and asp except in this song title. What I found, however, is the story of the turtle and the scorpion and not by Aesop, possibly even a bit older.

On this site they call it an old Zen tale:

A turtle was sitting beside a fast flowing river about to cross it,when a Scorpion approached him and asked to ride on his back to the other side.The Turtle was not too keen on the idea and said "you are very poisionous and could kill me with one bite".The Scorpion said "It is true,but I am a sensitive and caring creature and would not do that sort of thing".
"Very well " said the Turtle,"Lets go across".So they set off.
Halfway across the Scorpion stung the Turtle.The Turtle said "Why did you do that? Now we will both die."
The Scorpion replied--"It is in my nature".

This story must have been in Martin Simpson's mind when writing the song "The turtle and the asp".

Wolfgang