  Massive stars like company. However, low-mass companions have remained
extremely difficult to detect at angular separations ($\rho$) smaller than 1"
(~1000-3000 au considering typical distance to nearby massive stars) given the
large brightness contrast between the companion and the central star.
Constraints on the low-mass end of the companions mass-function for massive
stars are however needed, for example to help distinguishing between various
scenarios for the formation of massive stars. To obtain statistically
significant constraint on the presence of low-mass companions beyond the
typical detection limit of current surveys ($\Delta \mathrm{mag} \lesssim 5$ at
$\rho \lesssim 1$"), we initiated a survey of O and Wolf-Rayet stars in the
Carina region using the SPHERE coronagraphic instrument on the VLT. In this
first paper, we aim to introduce the survey, to present the methodology and to
demonstrate the capability of SPHERE for massive stars using the multiple
system QZ~Car. High-contrast imaging techniques, such as angular- and
spectral-differential imaging techniques as well as PSF-fitting, were applied
to detect and measure the relative flux of companions in each spectral channel
of the instrument. We detected 19 sources around the QZ~Car system with
detection limits of 9~mag at $\rho > 200$~mas for IFS and as faint as 13~mag at
$\rho$ > 1".8 for IRDIS (corresponding to sub-solar masses for potential
companions). All but two are reported here for the first time. Based on this
proof of concept, we showed that VLT/SPHERE allows us to reach the sub-solar
mass regime of the companion mass function. This paves the way for this type of
observation with a large sample of massive stars to provide novel constraints
on the multiplicity of massive stars in a region of the parameter space that
has remained inaccessible so far.
