The Higgs boson contributes to the part of a quark's mass that's affected by the quark's electroweak interactions. And that occurs through the specific couplings of a quark with the Higgs. The fact that a quark has three colors is crucial but enters in a more subtle fashion.
If you break down the mass of the top quark into two parts, it looks like
mt(pole)=mt(mt) + 4/3π mt(mt) αs(mt) + higher order
The first part is obtained from the Yukawa coupling of Higgs to top quark. The second part comes from QCD interactions. This is precisely because quark has three colors, i.e. it is not a color singlet. Because QCD coupling is quite large (αs ~ 0.12), color is confined, and also, a new piece is added to the top quark mass.
Ref: J. Gasser, H. Leutwyler, Phys.Rep. 87 (1982) 77-169
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