The nuclear spin and hfs in the $(5s5p)^{3}P_{1}$ state of 14-year ${\mathrm{Cd}}^{113m}$ have been measured by the optical double-resonance technique. The isotope was produced by the reaction ${\mathrm{Cd}}^{112}(n, \ensuremath{\gamma}){\mathrm{Cd}}^{113m}$. After irradiation, the ${\mathrm{Cd}}^{113m}$ was separated from the ${\mathrm{Cd}}^{112}$ with an electromagnetic mass separator. The nuclear spin $I$, and the hfs separations are: $I=\frac{11}{2}$; $\ensuremath{\nu}(\frac{13}{2}\ensuremath{-}\frac{11}{2})=4310.572(5)$ Mc/sec; $\ensuremath{\nu}(\frac{11}{2}\ensuremath{-}\frac{9}{2})=3949.625(7)$ Mc/sec. The hfs coupling constants, corrected to second order for interaction with the $^{3}P_{2}$ and $^{3}P_{0}$ states, are: $A(113m)=\ensuremath{-}686.0425(8)$ Mc/sec; $B(113m)=+169.047(9)$ Mc/sec. If we neglect nuclear structure and quadrupole shielding effects, the nuclear moments are: $\ensuremath{\mu}(113m)=\ensuremath{-}1.08885(13){\ensuremath{\mu}}_{N}$; $Q(113m)=\ensuremath{-}0.79(10)$ b. The ratio of the ${\mathrm{Cd}}^{113m}$ and ${\mathrm{Cd}}^{109}$ quadrupole moments is $\frac{Q(113m)}{Q(109)}=\ensuremath{-}1.02371(4)$; this result is independent of the shielding corrections. The spin and magnetic moment are consistent with a ${(2{d}_{\frac{5}{2}})}^{6}{(1{g}_{\frac{7}{2}})}^{8}{(1{h}_{\frac{11}{2}})}^{1}$ neutron assignment with some configuration mixing. We compare the observed quadrupole moments in cadmium and other spherical odd-neutron nuclei (A150, $189A210$) with predictions of the shell model, including configuration mixing, and with the quasiparticle model of Kisslinger and Sorensen. In addition, we show that the semiempirical formula $Q=\ensuremath{-}[\frac{(2j+1\ensuremath{-}2N)}{(2j+2)}]{Q}_{0}(Z)$ accurately predicts the quadrupole-moment ratios for the isotopes of several elements. We find that ${Q}_{0}(Z)$ exhibits strongly oscillatory dependence on the nuclear charge $Z$ with minima near the magic proton numbers. We suggest that ${Q}_{0}(Z)$ may be interpreted in terms of the quadrupolar polarizability of the proton core.