During the first observing run of LIGO, two gravitational wave events and one lower-significance trigger (LVT151012) were reported by the LIGO/Virgo collaboration. At the time of LVT151012, the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) was pointing at a region of the sky coincident with the high localization probability area of the event and thus permitted us to search for its electromagnetic counterpart (both prompt and afterglow emission). The imaging instruments on-board INTEGRAL (IBIS/ISGRI, IBIS/PICsIT, SPI, and the two JEM-X modules) have been exploited to attempt the detection of any electromagnetic emission associated with LVT151012 over 3 decades in energy (from 3 keV to 8 MeV). The omni-directional instruments on-board the satellite, i.e. the SPI-ACS and IBIS monitored the entire LVT151012 localization region at energies above 75 keV. We did not find any significant transient source that was spatially and/or temporally coincident with LVT151012, obtaining tight upper limits on the associated hard X-ray and $gamma$-ray radiation. For typical spectral models, the upper limits on the fluence of the emission from any 1 s long-lasting counterpart of LVT151012 ranges from $F_{gamma}=$3.5$times$10$^{-8}$ erg cm$^{-2}$ (20 - 200 keV) to $F_{gamma}$=7.1$times$10$^{-7}$ erg cm$^{-2}$ (75 - 2000 keV), constraining the ratio of the isotropic equivalent energy released in the electromagnetic emission to the total energy of the gravitational waves: $E_{75-2000~keV}/E_{GW}<$4.4$times$10$^{-5}$. Finally, we provide an exhaustive summary of the capabilities of all instruments on-board INTEGRAL to hunt for $gamma$-ray counterparts of gravitational wave events, exploiting both serendipitous and pointed follow-up observations. This will serve as a reference for all future searches.
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