The nuclear matrix elements for the spin operator and the momentum quadrupole operator are important for the interpretation of precision atomic physics experiments that search for violations of local Lorentz and CPT symmetry and for new spin-dependent forces. We use the configuration-interaction nuclear shell model and self-consistent mean-field theory to calculate the momentum matrix elements for ^{21}Ne, ^{23}Na, ^{133}Cs, ^{173}Yb, and ^{201}Hg. We show that these momentum matrix are strongly suppressed by the many-body correlations, in contrast to the well-known enhancement of the spatial quadrupole nuclear matrix elements.