  Quasi-one-dimensional (Q1D) Cr-based pnictide K$_2$Cr$_3$As$_3$ has aroused
great research interest due to its possible triplet superconducting pairing
symmetry. Recent experiments have shown that incorporating hydrogen atoms into
K$_2$Cr$_3$As$_3$ would significantly change its electronic and magnetic
properties. Hence, it's necessary to investigate the impact of hydrogen doping
in superconducting pairing symmetry of this material. Employing the hydrogen as
an non-trivial electron-doping, our calculates show that, different from the
$p_z$-wave obtained without hydrogen, the system exhibits $p_x\pm ip_y$ pairing
superconductivity under specific hydrogen doping. Specifically, we adopt the
random-phase-approximation approach based on a six-band tight-binding model
equipped with multi-orbital Hubbard interactions to study the hydrogen-doping
dependence of the pairing symmetry and superconducting $T_c$. Under the
rigid-band approximation, our pairing phase diagram shows the spin-triplet
pairing states is dominated through out the hydrogen-doping regime $x\in
(0,0.7)$. Particularly, the $T_c\sim x$ curve shows a peak at the 3D-quasi-1D
Lifshitz transition point, and the pairing symmetry around this doping level is
$p_x\pm ip_y$. The physical origin of this pairing symmetry is that the density
of states is mainly concentrated at $k_x(k_y)$ with large momentum. Due to the
three-dimensional character of the real material, this $p_x\pm ip_y$-wave
superconducting state possesses point gap nodes. We further provide experiment
prediction to identify this triplet $p_x\pm ip_y$-wave superconductivity.
