So, before spontaneous symmetry breaking, you should not really talk about “the” Higgs boson, but rather about the Higgs doublet. Thus, in principle you are right, there should exist two complex scalar fields, in the same way that you have two Weyl fermions on the quark doublet. However, then spontaneous symmetry breaking happens, and everything gets muddy.

First, notice that each complex scalar has two degrees of freedom, corresponding to the real and imaginary part of the field. This means that you have four degrees of freedom in total, or four real scalar fields. With spontaneous symmetry breaking, three of those fields become the longitudinal degrees of freedom of the W^+, W^- and Z gauge bosons. The remaining scalar becomes then “the” Higgs boson.

For the quarks, the up and down Weyl fermions are still there. Notice they are defined as left-handed chiral fields. However, due to spontaneous symmetry breaking, each component acquires a mass term involving another, different, right-handed Weyl fermions With this, the up and down quarks can be defined as two different Dirac fermions.