No. In general, there are no elliptic curves on an Abelian surface.
Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $\sqrt{-1}$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.