  The optical Ramsey technique is used to obtain precise measurements of the
hyperfine splittings in the $2\,^3\!S_1$ and $2\,^3\!P_J$ states of $^7$Li$^+$.
Together with bound-state quantum electrodynamic theory, the Zemach radius and
quadrupole moment of the $^7$Li nucleus are determined to be $3.35(1)$~fm and
$-3.86(5)$~fm$^2$ respectively, with the quadrupole moment deviating from the
recommended value of $-4.00(3)$~fm$^2$ by $1.75\sigma$. Furthermore, we
determine the quadrupole moment ratio of $^6$Li to $^7$Li as $0.101(13)$,
exhibiting a $6\sigma$ deviation from the previous measured value of
$0.020161(13)$ by LiF molecular spectroscopy. The results taken together
provide a sensitive test of nuclear structure models.
