  The statistical distribution and evolution of key properties (e.g. accretion
rate, mass, or spin) of active galactic nuclei (AGN), remain an open debate in
astrophysics. The ESA Euclid space mission, launched on July 1st 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 H$\alpha$+[NII] complex as Euclid is mainly focused on this emission
line as 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 H$\alpha$+[NII] is visible (0.89<z<1.83 at a line flux
$>2x10^{-16}$ erg s$^{-1}$ cm$^{-2}$, encompassing the peak of AGN activity at
$z\simeq 1-1.5$) within the spectral coverage of the red grism. Outside this
redshift range, the measurement efficiency decreases significantly. Overall, a
spectroscopic redshift is correctly determined for ~90% of type 2 AGN down to
an emission line flux of $3x10^{-16}$ erg s$^{-1}$ cm$^{-2}$, and for type 1
AGN down to $8.5x10^{-16}$ erg s$^{-1}$ cm$^{-2}$. Recovered black hole mass
values show a small offset with respect to the input values ~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 thousand AGN with homogeneous spectroscopic information.
