There's an interesting grey area...
According to the Wikipedia article on Henry S. Whitehead, "Williamson" was published posthumously in 1946 (he died in 1932). (He wrote/published his first story in 1924.) And he was a friend of Lovecraft's.
Another source cites the story as being written around 1910, but since it was never published in his lifetime, he would have had to share it directly with HPL for any influence to carry over.
It's almost as likely that there was some event around that time - some speculative article on evolution or something - that inspired both of them.