Not exactly.Her great grandfather used the spelling Alcox, his gravestone is in Edgewood Cemetery, Wolcott Conn. The stone reads:
Capt. JOHN ALCOXdied Sept. 27th 1808AEd 77 years Mrs. MARY his wifedied Feb. 28th 1807 AEd 71 years
It appears that her grandfather, Joseph Chatfield Alcox or Alcott is the one who changed the spelling. Her father's name, in everything we have seen is almost invariably spelled Alcott, though it is quite possible that he and his father changed it together while Amos was relatively young. (Joseph lived until Amos was thirty.)
Capt. John's is one of the earliest stones, most of the readable family stones are a bit later and have the name spelled Alcott. One is spelled Allcott.