  We compute the quadratic Euler characteristic of the symmetric powers of a
smooth, projective curve over any field $k$ that is not of characteristic two,
using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we
show over a field of characteristic zero that the power structure on the
Grothendieck-Witt ring introduced by Pajwani-P\'al computes the compactly
supported $\mathbb{A}^1$-Euler characteristic of symmetric powers for all
curves.
