This paper presents the results of a study on Coulomb drag resistivity in double bilayer graphene under the influence of an inhomogeneous substrate dielectric. Using the random phase approximation (RPA), we determine the system's polarization function and the frequency-dependent dielectric function, thereby calculating the Coulomb drag resistivity. The results indicate that the Coulomb drag resistivity increases with increasing temperature but decreases rapidly as the interlayer distance between the bilayer graphene sheets increases. Notably, when considering an inhomogeneous background dielectric, the Coulomb drag resistivity is significantly higher than that in the case of a homogeneous dielectric. This phenomenon arises from the modification of the Coulomb interaction potential between electrons in the two layers due to the inhomogeneity of the background dielectric. Furthermore, calculations reveal that at different temperatures, the Coulomb drag resistivity tends to decrease with increasing carrier density. However, a distinct difference emerges between two cases: small and large interlayer separations. When the interlayer sepaation is small, the Coulomb drag resistivity is more strongly influenced by carrier density and temperature. These findings provide further insights into the role of the inhomogeneous background dielectric in double bilayer graphene systems and suggest potential applications for next-generation graphene-based electronic devices.