Yes, the Higgs boson results in an attractive force between fermions given by the Yukawa potential (see here: https://en.wikipedia.org/wiki/Yukawa_interaction).

This potential takes basically the same form as the gravitational potential but with an additional exponential decay with distance due to the mass of the boson. Resulting in short-range interactions. So yes, in very short distances (~(inverse of the Higgs mass) ~ 10^-18 m) fermions do experience an attractive force between them due to the Higgs interaction.

It works in exactly the same manner the rest of interactions work, the boson (Higgs) is the mediator of the force.