A search for supersymmetric partners of gluons and quarks is presented, involving signatures with jets and either two isolated leptons (electrons or muons) with the same electric charge, or at least three isolated leptons. A data sample of proton-proton collisions at $s$ = 13 TeV recorded with the ATLAS detector at the Large Hadron Collider between 2015 and 2018, corresponding to a total integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, is used for the search. No significant excess over the Standard Model expectation is observed. The results are interpreted in simplified supersymmetric models featuring both R-parity conservation and R-parity violation, raising the exclusion limits beyond those of previous ATLAS searches to 1600 GeV for gluino masses and 750 GeV for bottom and top squark masses in these scenarios.