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Within an effective-field-theory framework, we present a model-independent analysis of the potential of discovering new physics by searching for lepton flavour violation in heavy quarkonium decays and, more in general, we study the phenomenology of lepton-flavour-violating (LFV) 2 quark - 2 lepton ($2q2\ell$) operators with two charm or bottom fields. We compute the constraints from LFV muon and tau decays on the new-physics operators that can induce LFV processes involving $c\bar c$ and $b\bar b$ systems, thus providing a comprehensive list of indirect upper limits on processes such as $J/ψ\to \ell\ell^\prime$, $Υ(nS) \to \ell\ell^\prime$, $Υ(nS) \to \ell\ell^\prime γ$ etc., which can be sought at BESIII, Belle II, and the proposed super tau-charm factory. We show that such indirect constraints are so stringent that they prevent the detection of quarkonium decays into $eμ$. In the case of decays of quarkonia into $\ellτ$ ($\ell=e,μ$), we find that an improvement by 2-3 orders of magnitude on the current sensitivities is in general required in order to discover or further constrain new physics. However, we show that cancellations among different contributions to the LFV tau decay rates are possible, such that $Υ(nS)\to \ellτ$ can saturate the present experimental bounds. We also find that, interestingly, searches for LFV $Z$ decays, $Z\to\ellτ$, at future $e^+e^-$ colliders are complementary probes of $2q2\ell$ operators with third generation quarks.