  We present results from a ~100 ks Chandra observation of the 2QZ Cluster
1004+00 structure at z = 2.23 (hereafter, 2QZ Clus). 2QZ Clus was originally
identified as an overdensity of four optically-selected QSOs at z = 2.23 within
a 15x15 arcmin^2 region. Narrow-band imaging in the near-IR revealed that the
structure contains an additional overdensity of 22 z = 2.23 Halpha-emitting
galaxies (HAEs), resulting in 23 unique z = 2.23 HAEs/QSOs. Our Chandra
observations reveal that 3 HAEs in addition to the 4 QSOs harbor powerfully
accreting supermassive black holes (SMBHs), with 2-10 keV luminosities of
~(8-60) x 10^43 ergs/s and X-ray spectral slopes consistent with unobscured
AGN. Using a large comparison sample of 210 HAEs in Chandra-COSMOS (C-COSMOS),
we find suggestive evidence that the AGN fraction increases with local HAE
galaxy density. The 2QZ Clus HAEs reside in a moderately overdense environment
(a factor of ~2 times over the field), and after excluding optically-selected
QSOs, we find the AGN fraction is a factor of ~3.5^+3.8_-2.2 times higher than
C-COSMOS HAEs in similar environments. Using stacking analyses of the Chandra
data and Herschel SPIRE observations at 250 um, we respectively estimate mean
SMBH accretion rates (Mdot) and star-formation rates (SFRs) for the 2QZ Clus
and C-COSMOS samples. We find that the mean 2QZ Clus HAE stacked 2-10 keV
luminosity is QSO-like (~6-10 x 10^43 ergs/s), and the implied Mdot/SFR \sim
(1.6-3.2) x10^-3 is broadly consistent with the local MBH/M* relation and z ~ 2
X-ray selected AGN. The C-COSMOS HAEs are on average an order of magnitude less
X-ray luminous and have Mdot/SFR \sim (0.2-0.4) x10^-3, comparable to z ~ 1-2
star-forming galaxies with similar mean X-ray luminosities. We estimate that a
periodic QSO phase with duty cycle ~2-8% would be sufficient to bring the
star-forming galaxies onto the local MBH/M* relation.
