No, adjoint functors are not related to the adjoint action of a group on its Lie algebra. They're related to another sort of "adjoint". Every linear map between vector spaces \$$T : V \to W \$$ gives rise to a linear map going the other way between the dual vector spaces, \$$T^{\ast} : W^{\ast} \to V^{\ast} \$$, and we call \$$T^{\ast} \$$ the **adjoint** of \$$T\$$.

Anyway, learn about left and right adjoints here and your world will expand in a very interesting way. All the stuff we're talking about here is much more generally applicable than differential geometry.