I'm not aware of 'alligator pear' being the name by which the avocado is known in the USA, and I couldn't find anything on the Internet to confirm that it was. 'Alligator pear' is just another name for the avocado.
CulinaryLore.com confirms this. It explains the origin of both names: 'avocado' comes from the Nahuatl word for a testicle; 'alligator pear' is because the fruit is pear–shaped, and its skin is rough like an alligator's.
Dictionary.com suggests that the 'alligator' element results partly from some mishearing of the word 'avocado' (presumably pronounced to rhyme with 'Play–Doh').
CulinaryLore concludes with the thought that "The term alligator pear was never favored by the industry [in the USA]."
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