Unimodular gravity (UG) is an interesting theory that may explain why the cosmological constant is extremely small, in contrast to general relativity (GR). The theory has only the transverse diffeomorphism invariance and this causes a lot of debate as to the equivalence of UG to GR in the covariant quantization. We study the covariant BRST quantization of UG by gauge fixing only the transverse diffeomorphism and show that the remaining physical degrees of freedom are two, the same number as GR. This is achieved by using antisymmetric tensor ghost fields which automatically satisfy the transverse condition without nonlocal projection operator. The theory exhibits the ghosts for ghosts phenomenon, which requires further gauge fixing and introduction of more ghosts. We identify the BRST quartet structure among the various fields and single out the remaining physical degrees of freedom.