The spin-lattice relaxation time for nuclei possessing electric quadrupole moments is determined mainly by the electric quadrupolar interactions between the nucleus and its environment. Here we give a microscopic formulation of the nuclear quadrupolar relaxation problem for a nucleus of a monatomic solute dissolved in molecular fluids. Our formulation is based on classical statistical mechanics and the interaction site model representation of the intermolecular potential. We assume that the fluctuating field gradient felt by the nucleus is caused mainly by the charge distribution of the surrounding solvent molecules, modulated by the Sternheimer (anti)shielding factor of the nucleus. In the extreme narrowing condition, the problem reduces to the determination of a time integral of the field gradient time correlation function G(t) on the nucleus position. By separation of G(t) into a static contribution G(t = 0) and a normalized time correlation function, we seek microscopic expressions for both G(t = 0) and its correlation time tau(Q). Within certain approximations we express tau(Q) in terms of the wavevector-dependent polarization charge correlation time tau(mu)(k), and G(t = 0) in terms of the pure solvent charge structure factor S-mu(k) and an analytical function of the solute cavity radius a. Taking as input tau(mu)(k) from molecular dynamics simulations of the pure solvent and S-mu(k) from the extended reference interaction-site model (XRISM) calculation, we apply the theory to the spin lattice relaxation rate of seven quadrupolar nuclei in acetonitrile solution. The solutes considered cover a wide range of size, charge, and nuclear spin quantum number. With reasonable choices of the solute cavity radii, the theory successfully reproduces the experimentally measured 1/T-1 for these solutes. Using molecular dynamics simulation, we also investigate the effects on 1/T-1 of neglecting the solute mobility. Our simulated data suggest that the solute mobility can reasonably be neglected for spin relaxation of heavy quadrupolar nuclei such as Kr and Xe. Finally, the dielectric continuum limit of our theory is discussed and compared with the related theory developed by Hynes and Wolynes.