In this paper, we prove fixed point theorems (FPTs) on multiplicative metric space (MMS) (X, ) by the help of integral-type contractions of self-quadruple mappings (SQMs), i.e., for ℘ 1 , ℘ 2 , ℘ 3 , ℘ 4 : X → R. For this, we assume that the SQMs are weakly compatible mappings and the pairs ℘ 1 , ℘ 3 and ℘ 2 , ℘ 4 satisfy the property (CLR ℘ 3 ℘ 4 ). Further, two corollaries are produced from our main theorem as special cases. The novelty of these results is that for the unique common fixed point (CFP) of the SQMs ℘ 1 , ℘ 2 , ℘ 3 , ℘ 4 , we do not need to the assumption of completeness of the MMS (X, ). These results generalize the work of Abdou, [A. A. N. Abdou, J. Nonlinear Sci. Appl., 9 (2016), 2244-2257, and many others in the available literature.