We present the first results of a study of the morphology, kinematics and ionization structure of infrared (IR) mergers/quasi-stellar objects (QSOs) with galactic winds. This study is based mainly on INTEGRAL two-dimensional (2D) fibre spectroscopy [obtained on the 4.2-m William Herschel Telescope (WHT), La Palma] combined with high-resolution Hubble Space Telescope (HST) observations. Clear evidence of outflow (OF) from the nucleus of the luminous infrared merger NGC 2623 is reported. Specifically: (i) the INTEGRAL 2D Hα, [N II] and [S II] emission line maps depict a cone-shaped extended nebula that emerges from the nucleus, with an aperture angle θ= 100°+/- 5° and reaching a distance of ~3.2 kpc from the nucleus; (ii) inside the nebula and in the central region, all the emission-line WHT spectra show low velocity blue/OF components, with = (-405 +/- 35) km s-1 and (iii) in the OF nebula, the emission line ratios are consistent with ionization by a dusty nuclear starburst plus shock heating. These results are consistent with a galactic wind process powered mainly by a nuclear starburst. The INTEGRAL 2D Hα and [N II]λ6583 velocity field (VF) maps for the main body of NGC 2623 (16.4 × 12.3 arcsec2; ~5.9 × 4.4 kpc2) show outflow motion in the nuclear and the Hα+[N II] nebular regions superposed on a general circular motion. This circular motion prevails inside r~ 1.5 kpc, and for larger radii we detected non-circular motions. In the central region, the average observed rotation curve was fitted with a model corresponding to a single-component Plummer spherical potential. After the subtraction of the Plummer and an axisymmetric polynomial model, the residues of the VF in both cases indicate ejection as the origin of the cone nebula. The fitted Plummer model implies a total mass of MT= 1.5 × 1010 Msolar and a spherical distribution of matter in the central region. The high-resolution HST WFPC2 F555W (~V) and F814W (~I) broad-band images display a strongly obscured nucleus in the apex of a small nuclear cone, an asymmetrical clumpy spiral arm located to the east of the nucleus, a ring plus an arc to the west and several large-scale filaments of dust. A good r1/4-law fit to the HST WFPC2 I band luminosity profile was found. In 85 per cent of the INTEGRAL 2D field we measure very high values (>1) of the [N II]λ6583/Hα and [S II]λ6717 + 31/Hα ratios, suggesting that shocks are important on large scales (in almost all the main body). Furthermore, the 2D full width at half maximum FWHM-[N II] and VF residual maps show a good spatial correlation, suggesting that the OF shocks ionize the gas and broaden the emission lines. However, close to the nucleus, the OF nebula shows low values of the [N II]λ6583/Hα ratio (in the range 0.1-0.4), indicating that photoionization by a dusty nuclear starburst also plays a significant role in the excitation of the nebula. The characteristics of the nucleus of NGC 2623 could be associated with a starburst-related LINER. The properties found in IR mergers/QSOs with galactic winds mainly underline the importance of studying the possible link between IR mergers with starburst + galactic wind -> IR QSOs with composite nature + galactic wind, and elliptical galaxies.