We consider a gravitational model in a Weyl-Cartan space-time, in which the Weitzenb\”{o}ck condition of the vanishing of the sum of the curvature and torsion scalar is also imposed. Moreover, a kinetic term for the torsion is also included in the gravitational action. The field equations of the model are obtained from a Hilbert-Einstein type variational principle, and they lead to a complete description of the gravitational field in terms of two fields, the Weyl vector and the torsion, respectively, defined in a curved background. The cosmological applications of the model are investigated for a particular choice of the free parameters in which the torsion vector is proportional to the Weyl vector. Depending on the numerical values of the parameters of the cosmological model, a large variety of dynamic evolutions can be obtained, ranging from inflationary/accelerated expansions to non-inflationary behaviors. In particular we show that a de Sitter type late time evolution can be naturally obtained from the field equations of the model. Therefore the present model leads to the possibility of a purely geometrical description of the dark energy, in which the late time acceleration of the Universe is determined by the intrinsic geometry of the space-time.