I found this:
Issue of April 10, 2007 -- Page 3
The use of "axe" as slang for a musical instrument dates back to 1955, i.e., in the edenic pre-Kiss days. The instrument to which "axe" was first applied, however, was not the guitar, but the saxophone. The logic may have been simply the "sax/axe" rhyme, but another theory ties "axe" to the "swing" of a jazz sax player in full stride. "Axe" was also later applied to the trumpet before becoming accepted as slang for the guitar, a use which has probably persisted in part because of the instrument's resemblance to an actual axe.