We used spatially resolved ALMA observations of the three CO isotopologues ^12^CO, ^13^CO and C^18^O (J=2-1) from CQ Tau to analyse the brightness temperature and kinematics of the gas disc. We detect significant spiral structures in both the brightness temperature and the rotation velocity of ^12^CO after subtraction of an azimuthally symmetric model, which may be tracing planet-disc interactions with an embedded planet or low-mass companion. The brightness temperature spirals are morphologically connected to spirals observed in NIR scattered light in the same disc, indicating a common origin. Together with the observed large dust and gas cavity, these spiral structures support the hypothesis of a massive embedded companion in the CQ Tau disc.