Because $ p(x) $ has integer coefficients (in particular, because it has rational coefficients), the other root of $ p(x) $ must be the radical conjugate of $ 4-\sqrt{11}, $ which is $ 4+\sqrt{11} $. Then, $ p(x) $ must take the form \[p(x) = A(x-(4-\sqrt{11}))(x-(4+\sqrt{11}))\]for some nonzero constant $ A $. This means that \[p(3) = A(-1+\sqrt{11})(-1-\sqrt{11}) = -10A\]and \[p(4) = A(\sqrt{11})(-\sqrt{11}) = -11A,\]so \[\frac{p(3)}{p(4)} = \frac{-10A}{-11A} = \boxed{\frac{10}{11}}.\]