Context. Both the stellar activity and the accretion processes of young stellar objects can induce variations in their radial velocity (RV). This variation is often modulated on the stellar rotation period and may hide a RV signal from a planetary or even a stellar companion. Aims. The aim of this study is to detect the companion of HQ Tau, the existence of which is suspected based on our previous study of this object. We also aim to derive the orbital elements of the system. Methods. We used multi-variate Gaussian process regression on the RV and the bisector inverse slope of a six-month high-resolution spectroscopic follow-up observation of the system to model the stellar activity. This allowed us to extract the Keplerian RV modulation induced by the suspected companion. Results. Our analysis yields the detection of a ∼50 M jup brown dwarf companion orbiting HQ Tau with a ∼126 day orbital period. Although this is consistent with the modulation seen on this dataset, it does not fit the measurements from our previous work three years earlier. In order to include these measurements in our analysis, we hypothesise the presence of a third component with orbital elements that are consistent with those of the secondary according to our previous analysis ( M B ∼ 48 M jup , P orb, B ∼ 126 days), and a ∼465 M jup tertiary with a ∼767 day orbital period. However, the hypothesis of a single companion with M B ∼ 188 M jup and P orb ∼ 247 days can fit both datasets and cannot be completely excluded at this stage of the analysis. Conclusions. At minima, HQ Tau is a single-lined spectroscopic binary, and several factors indicate that the companion is a brown dwarf and that a third component is responsible for larger RV variation on a longer timescale.