We investigate the prospects for discovering the Flavour Changing Neutral Current (FCNC) $tqZ$ couplings via two production processes yielding trilepton signals: top quark pair production $ppto tbar{t}$ with one top decaying to the $Z$ boson and one light jet and the anomalous single top plus $Z$ boson production process $ppto tZ$. We study these channels at various successors of the Large Hadron Collider~(LHC), i.e., the approved High-Luminosity LHC (HL-LHC) as well as the proposed High-Energy LHC~(HE-LHC) and Future Circular Collider in hadron-hadron mode (FCC-hh). We perform a full simulation for the signals and the relevant Standard Model (SM) backgrounds and obtain limits on the Branching Ratios (BRs) of $tto qZ~(q=u,c)$, eventually yielding a trilepton final state through the decay modes $tto b W^{+}to bell^{+} u_{ell}$ and $Zto ell^{+}ell^{-}$. The upper limits on these FCNC BRs at 95% Confidence Level (CL) are obtained at the HL-LHC with $sqrt s=14$ TeV and 3 ab$^{-1}$, at the HE-LHC with $sqrt s=27$ TeV and 15 ab$^{-1}$ as well as at the FCC-hh with $sqrt s=100$ TeV and 30 ab$^{-1}$.
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