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A Coulomb-excitation reorientation-effect measurement using the TIGRESS γ−ray spectrometer at the TRIUMF/ISAC II facility has permitted the determination of the 〈21+‖E2ˆ‖21+〉 diagonal matrix element in $^{12}$C from particle−γ coincidence data and state-of-the-art no-core shell model calculations of the nuclear polarizability. The nuclear polarizability for the ground and first-excited (21+) states in $^{12}$C have been calculated using chiral NN N$^{4}$LO500 and NN+3NF350 interactions, which show convergence and agreement with photo-absorption cross-section data. Predictions show a change in the nuclear polarizability with a substantial increase between the ground state and first excited 21+ state at 4.439 MeV. The polarizability of the 21+ state is introduced into the current and previous Coulomb-excitation reorientation-effect analyses of $^{12}$C. Spectroscopic quadrupole moments of QS(21+)=+0.053(44) eb and QS(21+)=+0.08(3) eb are determined, respectively, yielding a weighted average of QS(21+)=+0.071(25) eb, in agreement with recent ab initio calculations. The present measurement confirms that the 21+ state of $^{12}$C is oblate and emphasizes the important role played by the nuclear polarizability in Coulomb-excitation studies of light nuclei.