  The relativistic quantum dynamics of the generalized Klein-Gordon (KG)
oscillator having position-dependent mass in the G\"{o}del-type space-time is
investigated. We have presented the generalized KG oscillator in this
space-time, and discussed the effect of Cornell potential and linear potential
for our considered system. The modification from the parameters of
position-dependent mass and characterizing the space-time for the energy
spectrums are presented.
