We examine the H I gas kinematics of galaxy pairs in two clusters and a group using Australian Square Kilometre Array Pathfinder (ASKAP) WALLABY pilot survey observations. We compare the H I properties of galaxy pair candidates in the Hydra I and Norma clusters, and the NGC 4636 group, with those of non-paired control galaxies selected in the same fields. We perform H I profile decomposition of the sample galaxies using a tool, baygaud, which allows us to deblend a line-of-sight velocity profile with an optimal number of Gaussian components. We construct H I superprofiles of the sample galaxies via stacking of their line profiles after aligning the central velocities. We fit a double Gaussian model to the superprofiles and classify them as kinematically narrow and broad components with respect to their velocity dispersions. Additionally, we investigate the gravitational instability of H I gas discs of the sample galaxies using Toomre Q parameters and H I morphological disturbances. We investigate the effect of the cluster environment on the H I properties of galaxy pairs by dividing the cluster environment into three subcluster regions (i.e. outskirts, infalling, and central regions). We find that the denser cluster environment (i.e. infalling and central regions) is likely to impact the H I gas properties of galaxies in a way of decreasing the amplitude of the kinematically narrow H I gas (M^HI^_narrow_/M^HI^_total_), and increasing the Toomre Q values of the infalling and central galaxies. This tendency is likely to be more enhanced for galaxy pairs in the cluster environment.