  In this paper, we consider the possibility that Y(4260), Y(4360), $
\psi(4415) $ or X(4660) is the ground state or the first excited state of a
hybrid charmonium and examine whether $ \Upsilon ( 10860) $ is the ground state
or the first excited state of a hybrid bottomonium. Under a constituent quark
model, we carry out numerical calculations to obtain the spectra of both
electric gluon and magnetic gluon hybrid quarkonia. We see that the ground
state of a magnetic gluon hybrid charmonium and the first excited state of an
electric gluon hybrid are comparable in energy to $ \psi(4415) $ and that the
first excited states of an electric gluon hybrid bottomonium and the ground
state of a magnetic gluon hybrid appear a few hundred MeV above the mass of $
\Upsilon (10860 ) $. Also, we evaluate the widths of the decays of a hybrid
charmonium into $D^{(*)} \bar{D}^{(*) } $ and those of a hybrid bottomonium
into $ B^{(*) } \bar{B}^{(* ) } $. We show that if the exotic meson candidates
are hybrid charmonia, then they are magnetic gluon ones and argue that it seems
unlikely that $ \Upsilon (10860) $ is the ground state of an electric gluon
hybrid bottomonium.
