Flat spectrum radio quasars (FSRQs) constitute a class of jetted active galaxies characterized by a very luminous accretion disc, prominent and rapidly moving line-emitting cloud structures (broad-line region, BLR), and a surrounding dense dust structure known as dusty torus. The intense radiation field of the accretion disc strongly determines the observational properties of FSRQs. While hundreds of such sources have been detected at GeV energies, only a handful of them exhibit emission in the very-high-energy (VHE, E$\gtrsim 100$ GeV) range. This study presents the results and interpretation derived from a cumulative observation period of 174 h dedicated to nine FSRQs conducted with the Major Atmospheric Gamma-ray Imaging Cherenkov telescopes from 2008 to 2020. Our findings indicate no statistically significant ($\ge$5$\sigma$) signal for any of the studied sources, resulting in upper limits on the emission within the VHE energy range. In two of the sources, we derived quite stringent constraints on the gamma-ray emission in the form of upper limits. Our analysis focuses on modelling the VHE emission of these two sources in search for hints of absorption signatures within the BLR radiation field. For these particular sources, constraints on the distance between the emission region and the central black hole are derived using a phenomenological model. Subsequently, these constraints are tested using a framework based on a leptonic model.