The statistical distribution and evolution of key properties of active galactic nuclei (AGN), such as their accretion rate, mass, and spin, remains a subject of open debate in astrophysics. The ESA Euclid space mission, launched on July 1 2023, promises a breakthrough in this field. We create detailed mock catalogues of AGN spectra from the rest-frame near-infrared down to the ultraviolet - including emission lines - to simulate what Euclid will observe for both obscured (type 2) and unobscured (type 1) AGN. We concentrate on the red grisms of the NISP instrument, which will be used for the wide-field survey, opening a new window for spectroscopic AGN studies in the near-infrared. We quantify the efficiency in the redshift determination as well as in retrieving the emission line flux of the Hα+[N II] complex, as Euclid is mainly focused on this emission line, given that it is expected to be the brightest one in the probed redshift range. Spectroscopic redshifts are measured for 83% of the simulated AGN in the interval where the Hα is visible (i.e. 0.89 < z < 1.83 at a line flux of > 2 × 10−16 erg s−1 cm−2, encompassing the peak of AGN activity at z ≃ 1 − 1.5) within the spectral coverage of the red grism. Outside this redshift range, the measurement efficiency decreases significantly. Overall, a spectroscopic redshift iscorrectly determined for about 90% of type 2 AGN down to an emission line flux of roughly 3 × 10−16 erg s−1 cm−2, and for type 1 AGN down to 8.5 × 10−16 erg s−1 cm−2. Recovered values for black hole mass show a small offset with respect to the input values by about 10%, but the agreement is good overall. With such a high spectroscopic coverage at z < 2, we will be able to measure AGN demography, scaling relations, and clustering from the epoch of the peak of AGN activity down to the present-day Universe for hundreds of thousands of AGN with homogeneous spectroscopic information.