  We focus on Voisin's conjecture on 0-cycles on the self-product of surfaces
of geometric genus one, which arises in the context of the Bloch-Beilinson
filtration conjecture. We verify this conjecture for the family of Todorov
surfaces of type $(2,12)$, giving an explicit description of this family as
quotient surfaces of the complete intersection of four quadrics in
$\mathbb{P}^{6}$. We give some motivic applications.
