  In this paper we study quadratic points on the non-split Cartan modular
curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all
quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on
$X_{ns}^+(7)$. Using similar techniques for $p=11$, and employing a version of
Chabauty for symmetric powers of curves for $p=13$, we show that the same holds
for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain
classes of elliptic curves over quadratic fields are modular.
