According to Wikipedia, a harmonic function is one which satisfies: $$ \nabla^2 f = 0 $$ The spherical harmonics (also according to Wikipedia) satisfy the relation $$ \nabla^2 Y_l^m(\theta,\phi) = -\frac{l(l+1)}{r^2} Y_l^m(\theta,\phi) $$ which is 0 only if $l = 0$. So by this definition, are the spherical harmonic functions only harmonic when $l,m = 0$?

As a related question, if I change coordinates to some $\alpha(\theta,\phi), \beta(\theta,\phi)$ and I want to find the orthogonal functions similar to the spherical harmonics in the new $\alpha,\beta$ basis, how would I do that? Would I need to solve Laplace's equation $$ \nabla^2_{\alpha,\beta} f(\alpha,\beta) = 0 $$ where $\nabla^2_{\alpha,\beta}$ is the Laplacian expressed in the new coordinate system?