Contact pressure distribution throughout the contact interface has a vital role on the tribological aspects of the contact systems. Generally, contact of deformable bodies is a nonlinear problem. Viscoelastic materials have a time-dependent response, since both viscous and elastic characteristics depend on time. Such types of materials have the capability of storing and dissipating energy. When at least one of the contacting bodies is made of a viscoelastic material, contact problems become more difficult, and a nonlinear time-dependent contact problem is obtained. The objective of this paper is to develop an incremental adaptive computational model capable of handling quasistatic viscoelastic frictionless contact problems. The Wiechert model, as an effective model capable of describing both creep and relaxation phenomena, is adopted to simulate the linear behavior of viscoelastic materials. The resulting constitutive integral equations are linearized and, therefore, complications that arise during the direct integration of these equations, specially with contact problems, are avoided. In addition, the incremental convex programming method is adopted and modified to accommodate the contact problem of viscoelastic bodies. The Lagrange multiplier method is adopted to enforce the contact constraints. Two different contact problems are presented to demonstrate the efficient applicability of the proposed model.