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We study the effect of two Modified Gravity (MG) theories, $f(R)$ and nDGP, on three probes of large-scale structure, the real space power spectrum estimator $Q_0$, bispectrum and voids, and validate fast approximate COLA simulations against full $N$-body simulations for the prediction of these probes. We find that using the first three even multipoles of the redshift space power spectrum to estimate $Q_0$ is enough to reproduce the MG boost factors of the real space power spectrum for both halo and galaxy catalogues. By analysing the bispectrum and reduced bispectrum of Dark Matter (DM), we show that the strong MG signal present in the DM bispectrum is mainly due to the enhanced power spectrum. We warn about adopting screening approximations in simulations as this neglects non-linear contributions that can source a significant component of the MG bispectrum signal at the DM level, but we argue that this is not a problem for the bispectrum of galaxies in redshift space where the signal is dominated by the non-linear galaxy bias. Finally, we perform void-finding on our galaxy mock catalogues by the ZOBOV watershed algorithm. To apply a linear model for Redshift-Space Distortion (RSD) in the void-galaxy cross-correlation function, we first examine the effects of MG on the void profiles entering into the RSD model. We find relevant MG signals in the integrated-density, velocity dispersion and radial velocity profiles in the nDGP theory. Fitting the RSD model for the linear growth rate, we recover the linear theory prediction in an nDGP model, which is larger than the $Λ$CDM prediction at the $3 σ$ level. In $f(R)$ theory we cannot naively compare the results of the fit with the linear theory prediction as this is scale-dependent, but we obtain results that are consistent with the $Λ$CDM prediction.