Context. Swift J1749.4-2807 is a transient accreting millisecond X-ray pulsars and the first to display X-ray eclipses. It therefore holds a strong potential for accurate mass measurements in a low-mass X-ray binary system. Aims. Determining the companion star's radial velocity would make it possible to fully resolve the system and to accurately measure the mass of the neutron star based on dynamical measurements. Unfortunately, no optical or near infrared (NIR) counterpart has been identified to date for this system, whether in outburst or in quiescence.Methods. We performed a photometric study of the field of Swift J1749.4-2807 during quiescence to search for a variable counterpart. The source direction lies on the Galactic plane, making any search for its optical/NIR counterpart challenging. To minimize the effects of field crowding and interstellar extinction, we carried out our observations using the adaptive optics NIR imager NACO mounted at the ESO Very Large Telescope (VLT).Results. From the analysis of public Swift X-ray data obtained during outburst, we derived the most precise (1.6 radius) position for this source. Due to the extreme stellar crowding of the field, 41 sources are detected in our VLT images within the X-ray error circle, with some of them possibly showing variability consistent with the expectations. Conclusions. We carried out the first deep imaging campaign devoted to the search for the quiescent NIR counterpart of Swift J1749.4-2807. Our results allow us to provide constraints on the nature of the companion star of this system. Furthermore, they suggest that future phase-resolved NIR observations (performed with large-aperture telescopes and adaptive optics) covering the full orbital period of the system are likely to identify the quiescent counterpart of Swift J1749.4-2807, through measuring its orbital variability, opening the possibility of dynamical studies of this unique source.