This Letter presents a search for quantum black-hole production using $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected with the ATLAS detector in $pp$ collisions at the LHC at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$. The quantum black holes are assumed to decay into a final state characterized by a lepton (electron or muon) and a jet. In either channel, no event with a lepton-jet invariant mass of 3.5 TeV or more is observed, consistent with the expected background. Limits are set on the product of cross sections and branching fractions for the $\text{lepton}+\text{jet}$ final states of quantum black holes produced in a search region for invariant masses above 1 TeV. The combined 95% confidence level upper limit on this product for quantum black holes with threshold mass above 3.5 TeV is 0.18 fb. This limit constrains the threshold quantum black-hole mass to be above 5.3 TeV in the model considered.