We construct a quark-soliton model of the nucleon which interpolates between the MIT bag model of arbitrarily confined relativistic quarks and the Skyrme model in which the nucleon is a topological knot in the pion field. Confinement is achieved using the color dielectric model. The field equations are solved numerically for the so called hedgehog state for which the field equations reduce to radial ones. Nucleon observables are calculated by projecting the quark-soliton hedgehog ground state onto spin-isospin eigenstates, which are taken to be variationally best nucleon and nucleon isobar states.