  The Dark Energy Spectroscopic Instrument (DESI) cosmology survey includes a
Bright Galaxy Survey (BGS) which will yield spectra for over ten million bright
galaxies (r<20.2 AB mag). The resulting sample will be valuable for both
cosmological and astrophysical studies. However, the star/galaxy separation
criterion implemented in the nominal BGS target selection algorithm excludes
quasar host galaxies in addition to bona fide stars. While this excluded
population is comparatively rare (~3-4 per square degrees), it may hold
interesting clues regarding galaxy and quasar physics. Therefore, we present a
target selection strategy that was implemented to recover these missing active
galactic nuclei (AGN) from the BGS sample. The design of the selection criteria
was both motivated and confirmed using spectroscopy. The resulting BGS-AGN
sample is uniformly distributed over the entire DESI footprint. According to
DESI survey validation data, the sample comprises 93% quasi-stellar objects
(QSOs), 3% narrow-line AGN or blazars with a galaxy contamination rate of 2%
and a stellar contamination rate of 2%. Peaking around redshift z=0.5, the
BGS-AGN sample is intermediary between quasars from the rest of the BGS and
those from the DESI QSO sample in terms of redshifts and AGN luminosities. The
stacked spectrum is nearly identical to that of the DESI QSO targets,
confirming that the sample is dominated by quasars. We highlight interesting
small populations reaching z>2 which are either faint quasars with nearby
projected companions or very bright quasars with strong absorption features
including the Lyman-apha forest, metal absorbers and/or broad absorption lines.
