We study the stellar and gas kinematics of the brightest group galaxies (BGGs) in dynamically relaxed and unrelaxed galaxy groups for a sample of 154 galaxies in the SAMI galaxy survey. We characterize the dynamical state of the groups using the luminosity gap between the two most luminous galaxies and the BGG offset from the luminosity centroid of the group. We find that the misalignment between the rotation axis of gas and stellar components is more frequent in the BGGs in unrelaxed groups, although with quite low statistical significance. Meanwhile, galaxies whose stellar dynamics would be classified as "regular rotators"based on their kinemetry are more common in relaxed groups. We confirm that this dependency on group dynamical state remains valid at fixed stellar mass and Sérsic index. The observed trend could potentially originate from a differing BGG accretion history in virialized and evolving groups. Among the halo relaxation probes, the group BGG offset appears to play a stronger role than the luminosity gap on the stellar kinematic differences of the BGGs. However, both the group BGG offset and luminosity gap appear to roughly equally drive the misalignment between the gas and stellar component of the BGGs in one direction. This study offers the first evidence that the dynamical state of galaxy groups may influence the BGG's stellar and gas kinematics, and we call for further studies using a larger sample with higher signal-to-noise.