  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 $J/\psi \to \ell\ell^\prime$,
$\Upsilon(nS) \to \ell\ell^\prime$, $\Upsilon(nS) \to \ell\ell^\prime \gamma$
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 $e\mu$. In the case of decays
of quarkonia into $\ell\tau$ ($\ell=e,\mu$), 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 $\Upsilon(nS)\to \ell\tau$ can saturate the present
experimental bounds. We also find that, interestingly, searches for LFV $Z$
decays, $Z\to\ell\tau$, at future $e^+e^-$ colliders are complementary probes
of $2q2\ell$ operators with third generation quarks.
