  Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the
smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$
is Koszul by constructing a quadratic Gr\"obner basis for the associated toric
ideal. Our proofs use the construction of graphs corresponding to fibers of the
toric map. As a consequence, we conclude that the Rees algebra is also Koszul.
