Irregularity from a smooth Albanese curve image

If the image of the Albanese morphism of a surface is a smooth curve $C$ of genus $g$, then $q(X)=g$. Pullback of one-forms from $C$ gives $g\le q(X)$, while the universal property gives a surjection from the Jacobian of $C$ onto $\mathrm{Alb}(X)$ and hence $q(X)\le g$.