ABSTRACT We investigate the role of galaxy mergers on supermassive black hole (SMBH) accretion and star formation quenching in three state-of-the-art cosmological simulations with contrasting physics models: EAGLE, Illustris, and IllustrisTNG (TNG). We find that recently coalesced ‘post-mergers’ in all three simulations have elevated SMBH accretion rates by factors of ∼2–5. However, rapid (within 500 Myr of coalescence) quenching of star formation is rare, with incidence rates of $0.4{{\ \rm per\ cent}}$ in Illustris, $4.5{{\ \rm per\ cent}}$ in EAGLE, and $10{{\ \rm per\ cent}}$ in TNG. The rarity of quenching in post-mergers results from substantial gas reservoirs that remain intact after the merger. The post-mergers that do successfully quench tend to be those that had both low pre-merger gas fractions as well as those that experience the largest gas losses. Although rare, the recently quenched fraction of post-mergers is still elevated compared to a control sample of non-mergers by factors of two in TNG and 11 in EAGLE. Conversely, quenching is rarer in Illustris post-mergers than in their control. Recent observational results by Ellison et al. have found rapid quenching to be at least 30 times more common in post-mergers, a significantly higher excess than found in any of the simulations. Our results therefore indicate that whilst merger-induced SMBH accretion is a widespread prediction of the simulation, its link to quenching depends sensitively on the physics models, and that none of the subgrid models of the simulations studied here can fully capture the connection between mergers and rapid quenching seen in observations.