A new interior-exterior penalty method for solving quasi-variational inequality and pseudo-monotone operator arising in two-dimensional point contact problem is analyzed and developed in discontinuous Galerkin finite volume (DG-FVEM) framework. We derive a discrete DG-FVEM formulation of the problem and prove existence and uniqueness results for it. Optimal error estimates in $H^1$ and $L^2$ norm are derived under a light load parameter assumptions. In addition, the article provides a complete algorithm to tackle all numerical complexities appear in the solution procedure. Numerical outcomes are presented for light, moderate and relative high load conditions. The variations of load parameter and its effect on the evolution of deformations and pressure profile are evaluated and described. This method is well suited for solving elasto-hydrodynamic lubrication point contact problems and can probably be treated as commercial software. Furthermore, the results give a hope for the further development of the scheme for extreme load condition, observations in a more realistic operating situation which will be discussed in part II.