Marcel Berger, on p. 696 of his Springer-book: A panoramic view of Riemannian Geometry, writes "Next we define the canonical musical duality", referring to a footnote: "These dualities are called musical because they are often written with symbols like ♭: V $\rightarrow$ V* and ♯: V* $\rightarrow$ V."
Even if this is not the first metioning of "musical", we can assume that Berger invented that expression, because in a panoramic viev of over 800 pages, a reference to another originator could have been expected.
Further Rob Kusner's hint here # or "sharp" is what Marcel Berger calls a "musical isomorphism" and Olivier Bégassat's comment here seem to provide sufficient evidence to settle this question.