We calculate the tensor charge of the quark in the QCD-like theory in the Landau gauge using the Schwinger-Dyson formalism. It is found that the dressed tensor charge of the quark is significantly suppressed against the bare quark contribution, and the result agrees qualitatively with the analyses in the collinear factorization approach and lattice QCD. We also analyze the quark confinement effect with the phenomenological strong coupling given by Richardson and find that this contribution is small. We show that the suppression of the quark tensor charge is due to the superposition of the spin flip of the quark arising from the successive emission of gluons that dress the tensor vertex. We also consider the relation between the quark and the nucleon electric dipole moments by combining with the simple constituent quark model.