Let $ p(x) = ax^2 + bx + c $. Then \begin{align*} 7a + b \sqrt{7} + c &= 22, \\ 11a + b \sqrt{11} + c &= 30.\end{align*}Since $ a, $ $ b, $ and $ c $ are rational, the only way that the equations above can hold is if $ b = 0 $. Then \begin{align*} 7a + c &= 22, \\ 11a + c &= 30.\end{align*}Solving this system, we find $ a = 2 $ and $ c = 8 $. Then $ p(x) = 2x^2 + 8, $ so $ p(\sqrt{17}) = 2 \cdot 17 + 8 = \boxed{42} $.