  A search for vectorlike quarks is presented, which targets their decay into a
$Z$ boson and a third-generation Standard Model quark. In the case of a
vectorlike quark $T$ ($B$) with charge $+2/3e$ ($-1/3e$), the decay searched
for is $T \rightarrow Zt$ ($B \rightarrow Zb$). Data for this analysis were
taken during 2015 and 2016 with the ATLAS detector at the Large Hadron Collider
and correspond to an integrated luminosity of 36.1 fb$^{-1}$ of $pp$ collisions
at $\sqrt{s} = 13$ TeV. The final state used is characterized by the presence
of $b$-tagged jets, as well as a $Z$ boson with high transverse momentum, which
is reconstructed from a pair of opposite-sign same-flavor leptons. Pair and
single production of vectorlike quarks are both taken into account and are each
searched for using optimized dileptonic exclusive and trileptonic inclusive
event selections. In these selections, the high scalar sum of jet transverse
momenta, the presence of high-transverse-momentum large-radius jets, as well as
- in the case of the single-production selections - the presence of forward
jets are used. No significant excess over the background-only hypothesis is
found and exclusion limits at 95% confidence level allow masses of vectorlike
quarks of $m_T > 1030$ GeV ($m_T > 1210$ GeV) and $m_B > 1010$ GeV ($m_B >
1140$ GeV) in the singlet (doublet) model. In the case of 100% branching ratio
for $T\rightarrow Zt$ ($B\rightarrow Zb$), the limits are $m_T > 1340$ GeV
($m_B > 1220$ GeV). Limits at 95% confidence level are also set on the coupling
to Standard Model quarks for given vectorlike quark masses.
