  This study explores the cosmological implications of the $F(R, T)$
quasi-dilaton massive gravity theory, a modification of the de
Rham-Gabadadze-Tolley (dRGT) massive gravity theory. Our analysis focuses on
the self-accelerating solution of the background equations of motion, which are
shown to exist in the theory. Notably, we find that the theory features an
effective cosmological constant, which has important implications for our
understanding of the universe's accelerated expansion. To test the viability of
the $F(R, T)$ quasi-dilaton massive gravity theory, we utilize the Union2 Type
Ia Supernovae (SNIa) dataset, comprising 557 observations. Our results
demonstrate that the theory is capable of explaining the accelerated expansion
of the universe without requiring the presence of dark energy. This finding
supports the potential of the $F(R, T)$ quasi-dilaton massive gravity theory as
an alternative explanation for the observed cosmic acceleration. Moreover, we
investigate the properties of tensor perturbations within the framework of this
theory and derive a novel expression for the dispersion relation of
gravitational waves. Our analysis reveals interesting features of the modified
dispersion relation in the Friedmann-Lema\^itre-Robertson-Walker cosmology,
providing new insights into the nature of gravitational waves in the contest of
the $F(R, T)$ quasi-dilaton massive gravity theory.
