In the present paper, the fundamental problem of calculating the electric field gradient (EFG) experienced by a highly idealized solute, represented by a general point quadrupole immersed in an anisotropic uniaxial medium, has been tackled. Following a generalized reaction field approach (based upon the original ideas and the "mean-field philosophy" due to Kirkwood and Onsager) in the linear response approximation, a closed analytical expression of the EFG has been derived (to the best of our knowledge, for the first time). The obtained expression is particularly simple and elegant, also thanks to the oversimplifying approximation that the Virtual cavity containing the solute is assumed to be perfectly spherical. This compact and manageable formula, obtained by a rigorous mathematical derivation (unlike other mean-field phenomenological models previously Suggested in literature) can be useful to investigate and better understand a likely orientational mechanism, partly responsible for the ordering of small Solutes dissolved in nematic mesophases, based on the interaction between the electric quadrupole of the solute and the electric field gradient of the anisotropic uniaxial medium (in the next paper of this issue, the formulation obtained in this work is widely tested oil a variety of uniaxial and biaxial Solutes dissolved in different nematic solvents).