A search is described for the production of a pair of bottom-type vectorlike quarks (VLQs), each decaying into a $b$ or $\overline{b}$ quark and either a Higgs or a $Z$ boson, with a mass greater than 1000 GeV. The analysis is based on data from proton-proton collisions at a 13 TeV center-of-mass energy recorded at the CERN LHC, corresponding to a total integrated luminosity of $137\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. As the predominant decay modes of the Higgs and $Z$ bosons are to a pair of quarks, the analysis focuses on final states consisting of jets resulting from the six quarks produced in the events. Since the two jets produced in the decay of a highly Lorentz-boosted Higgs or $Z$ boson can merge to form a single jet, nine independent analyses are performed, categorized by the number of observed jets and the reconstructed event mode. No signal in excess of the expected background is observed. Lower limits are set on the VLQ mass at 95% confidence level equal to 1570 GeV in the case where the VLQ decays exclusively to a $b$ quark and a Higgs boson, 1390 GeV for when it decays exclusively to a $b$ quark and a $Z$ boson, and 1450 GeV for when it decays equally in these two modes. These limits represent significant improvements over the previously published VLQ limits.