  The supersymmetric index of the 4d $\mathcal{N} = 1$ theory realized by a
brane tiling coincides with the partition function of an integrable 2d lattice
model. We argue that a class of half-BPS surface defects in brane tiling models
are represented on the lattice model side by transfer matrices constructed from
L-operators. For the simplest surface defects in theories with $\mathrm{SU}(2)$
flavor groups, we identify the relevant L-operator as that discovered by
Sklyanin in the context of the eight-vertex model. We verify this
identification by computing the indices of class-$\mathcal{S}$ and
-$\mathcal{S}_k$ theories in the presence of the surface defects.
