  We present Floquet calculations of high harmonic generation (HHG) for the
lowest two electronic states of the $\mbox{H}_2^+$ ion by strong
continuous-wave laser fields. We solve the non-Hermitian matrix problem to get
accurate solutions to the periodic time-dependent Schr\"odinger equation (TDSE)
by applying a pseudospectral representation combined with a complex absorbing
potential method. This represents an alternative approach to direct TDSE
solutions to obtain the harmonic spectra for the ion. We compare our HHG rates
for the lower and upper states of $\mbox{H}^{+}_{2}$, which correspond to the
gerade and ungerade ground states in the field-free case, with previously
obtained results in the literature. We show that the enhancement of the
ionization rates at the critical internuclear separation $R_{c}\approx 8\,au$
plays some role in the appearance of very strong harmonic orders $n=5-11$ at
$\lambda = 1064\,nm$ and $n=5-9$ at $\lambda = 800\,nm$ and intensity
$I=10^{14}\,W/cm^{2}$.
