Accurate prediction of electromagnetic moments in odd-A nuclei near doubly magic cores is challenging due to their sensitivity to core polarization and single-particle structure. This study investigates the predictive performance of Skyrme-HFB (SLy4) against shell model (SM) calculations for quadrupole moment (Q₂₀) and magnetic dipole (M₁₀) in the selected nuclei (¹⁷O, ¹⁷F, ³⁹Ca, ³⁹K, ⁴⁷Ca, ⁴⁹Sc, ¹³¹In, ¹³³Sb, ¹³³Sn, and ²⁰⁹Bi) near doubly magic cores using the Hartree-Fock-Bogoliubov (HFB) method and the SM method. The SLy4 interaction was used in HFB calculations. On the other hand, SM calculations were performed in sd, sd-pf, fp, and jj shells using USDC, SDPF-U, GXPF1A, jj45pn, and sn100pn interactions. The calculated results of both methods were compared with experimental data from the IAEA/INDC nuclear datasets. For Q₂₀ Skyrme-HFB predicted the correct sign with minor deviations in the value of light nuclei (e.g., ¹⁷O, ¹⁷F, ³⁹Ca, ⁴⁷Ca) by about −7% to +6%, but for heavier nuclei like ³⁹K and ⁴⁹Sc the deviation is much higher about (-88% and -85.7%), and for nuclei around ¹³²Sn and ²⁰⁸Pb cores (¹³¹In, ¹³³Sb, ¹³³Sn, ²⁰⁹Bi) the deviation was also high, ranging from -81% to -96%. Meanwhile, the SM Q₂₀ results showed a significant improvement, with a deviation of 1-16% for the whole set. For M₁₀, HFB calculations were close to experiment for light nuclei (¹⁷O and ¹⁷F) with deviation (0.99% and 1.53%), respectively, and reasonable for nuclei like (⁴⁷Ca ≈ 4.5% and ³⁹Ca ≈ 11.5%), but deviated strongly for ³⁹K by -73.87%. M₁₀ for heavier nuclei was not evaluated within HFB in the present work. SM calculation reproduced M₁₀ with absolute percent deviations of  across the studied nuclei (median ), where the lowest  values were ¹⁷F (0.03%), ²⁰⁹Bi (0.29%), ³⁹K (0.38%), and ¹⁷O (0.87%), while the highest  values were ³⁹Ca (29.23%), ¹³³Sn (25.53%), and ¹³³Sb (23.33%).