  The production of the heavy $(c\bar{c})$-quarkonium, $(c\bar{b})$-quarkonium
and $(b\bar{b})$-quarkonium states ($(Q\bar{Q'})$-quarkonium for short), via
the $W^+$ semi-inclusive decays, has been systematically studied within the
framework of the non-relativistic QCD. In addition to the two color-singlet
$S$-wave states, we also discuss the production of the four color-singlet
$P$-wave states $|(Q\bar{Q'})(^1P_1)_{\bf 1}>$ and $(Q\bar{Q'})(^3P_J)_{\bf
1}>$ (with $J=(1,2,3)$) together with the two color-octet components
$|(Q\bar{Q'})(^1S_0)_{\bf 8}>$ and $|(Q\bar{Q'})(^3S_1)_{\bf 8}>$. Improved
trace technology is adopted to derive the simplified analytic expressions at
the amplitude level, which shall be useful for dealing with the following
cascade decay channels. At the LHC with the luminosity ${\cal L}\propto
10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$ TeV, sizable
heavy-quarkonium events can be produced through the $W^+$ boson decays, i.e.
$2.57\times10^6$ $\eta_c$, $2.65\times10^6$ $J/\Psi$ and $2.40\times10^6$
$P$-wave charmonium events per year can be obtained; and $1.01\times10^5$
$B_c$, $9.11\times10^4$ $B^*_c$ and $3.16\times10^4$ $P$-wave
$(c\bar{b})$-quarkonium events per year can be obtained. Main theoretical
uncertainties have also been discussed. By adding the uncertainties caused by
the quark masses in quadrature, we obtain $\Gamma_{W^+\to (c\bar{c})+c\bar{s}}
=524.8^{+396.3}_{-258.4}$ KeV, $\Gamma_{W^+\to (c\bar{b})+b\bar{s}}
=13.5^{+4.73}_{-3.29}$ KeV, $\Gamma_{W^+\to (c\bar{b})+c\bar{c}}=
1.74^{+1.98}_{-0.73}$ KeV and $\Gamma_{W^+\to (b\bar{b})+c\bar{b}}=
38.6^{+13.4}_{-9.69}$ eV.
