  The celebrated Kitaev honeycomb model provides an analytically tractable
example with an exact quantum spin liquid ground state. While in real
materials, other types of interactions besides the Kitaev coupling ($K$) are
present, such as the Heisenberg ($J$) and symmetric off-diagonal ($\Gamma$)
terms, and these interactions can also be generalized to a triangular lattice.
Here, we carry out a comprehensive study of the $J$-$K$-$\Gamma$ model on the
triangular lattice covering the full parameters region, using the combination
of the exact diagonalization, classical Monte Carlo and analytic methods. In
the HK limit ($\Gamma=0$), we find five quantum phases which are quite similar
to their classical counterparts. Among them, the stripe-A and dual N\'{e}el
phase are robust against the $\Gamma$ term, in particular the stripe-A extends
to the region connecting the $K=-1$ and $K=1$ for $\Gamma<0$. Though the
120$^\circ$ N\'{e}el phase also extends to a finite $\Gamma$, its region has
been largely reduced compared to the previous classical result. Interestingly,
the ferromagnetic (dubbed as FM-A) phase and the stripe-B phase are unstable in
response to an infinitesimal $\Gamma$ interaction. Moreover, we find five new
phases for $\Gamma\ne 0$ which are elaborated by both the quantum and classical
numerical methods. Part of the space previously identified as 120$^\circ$
N\'{e}el phase in the classical study is found to give way to the modulated
stripe phase. Depending on the sign of the $\Gamma$, the FM-A phase transits
into the FM-B ($\Gamma>0$) and FM-C ($\Gamma<0$) phase with different spin
orientations, and the stripe-B phase transits into the stripe-C ($\Gamma>0$)
and stripe-A ($\Gamma<0$). Around the positive $\Gamma$ point, due to the
interplay of the Heisenberg, Kiatev and $\Gamma$ interactions, we find a
possible quantum spin liquid with a continuum in spin excitations.
