  We construct minimum effective models to investigate the pairing symmetry in
the newly discovered quasi-one-dimensional superconductor K$_2$Cr$_3$As$_3$. We
show that a minimum three-band model based on the $d_{z^2}$, $d_{xy}$ and
$d_{x^2-y^2}$ orbitals of one Cr sublattice can capture the band structures
near Fermi surfaces. In both weak and strong coupling limits, the standard
random phase approximation (RPA) and mean-field solutions consistently yield
the triplet $p_z$-wave pairing as the leading pairing symmetry for physically
realistic parameters. The triplet pairing is driven by the ferromagnetic
fluctuations within the sublattice. The gap function of the pairing state
possesses line gap nodes on the $k_z=0$ plane on the Fermi surfaces.
