A search is performed for the process $pp \to G^* \to B_H\bar b/\bar B_H b \to H b \bar b \to b\bar bb\bar b$, predicted in composite Higgs scenarios, where $G^*$ is a heavy colour octet vector resonance and $B_H$ a vector-like quark of charge $-1/3$. The data were obtained from $pp$ collisions at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 19.5 fb$^{-1}$, recorded by the ATLAS detector at the LHC. The largest background, multijet production, is estimated using a data-driven method. No significant excess of events with respect to Standard Model predictions is observed, and upper limits on the production cross section times branching ratio are set. Comparisons to the predictions from a specific benchmark model are made, resulting in lower mass limits in the two-dimensional mass plane of $m_{G^*}$ vs. $m_{B_H}$.