Uniform electric field quadrupole polarizabilities and shielding factors are calculated for a large number of two- to twenty-electron S-state atoms and ions. These coefficients, relating the induced quadrupole distortion to the external field strength, are calculated using uniform-field-perturbed analytical Hartree-Fock functions. The resulting polarizabilities are large in some cases, the values varying by many orders of magnitude through an isoelectronic series. Specifically, for field strenth approximately 106 V/cm, the induced moments for negative ions, neutrals, and some positive ions are of the same order of magnitude as permanent molecular quadrupole moments, and therefore the effect should be measurable by methods suggested in the literature. It is explicitly demonstrated that these polarizabilities and shielding factors are independent of the existence of a gradient in the external field. Thus, the coefficients are used to compare the quadrupole distortion induced by a uniform field with that induced by a field gradient through the usual (field-gradient) quadrupole polarizability. In particular, for fields and gradients of the order of magnitude of those expected in alkali halide molecules, on the basis of a point-charge model, the uniform-field contribution to the induced quadrupole in the halide ions is of the same order of magnitude as the gradient-induced contribution, in agreement with a previous result of Foley, Sternheimer, and Tycko. Finally, the sensitivity of the calculated polarizabilities and shielding factors to the quality of the zeroth-order Hartree-Fock function is investigated.