  Aiming at exploring the nature of dark energy, we use thirty-six
observational Hubble parameter data (OHD) in the redshift range $0 \leqslant z
\leqslant 2.36$ to make a cosmological model-independent test of the two-point
$Omh^2(z_{2};z_{1})$ diagnostic. In $\Lambda$CDM, we have $Omh^2 \equiv
\Omega_{m}h^2$, where $\Omega_{m}$ is the matter density parameter at present.
We bin all the OHD into four data points to mitigate the observational
contaminations. By comparing with the value of $\Omega_{m}h^2$ which is
constrained tightly by the Planck observations, our results show that in all
six testing pairs of $Omh^2$ there are two testing pairs are consistent with
$\Lambda$CDM at $1\sigma$ confidence level (CL), whereas for another two of
them $\Lambda$CDM can only be accommodated at $2\sigma$ CL. Particularly, for
remaining two pairs, $\Lambda$CDM is not compatible even at $2\sigma$ CL.
Therefore it is reasonable that although deviations from $\Lambda$CDM exist for
some pairs, cautiously, we cannot rule out the validity of $\Lambda$CDM. We
further apply two methods to derive the value of Hubble constant $H_0$
utilizing the two-point $Omh^2(z_{2};z_{1})$ diagnostic. We obtain $H_0 =
71.23\pm1.54$ ${\mathrm{km \ s^{-1} \ Mpc^{-1}}}$ from inverse variance
weighted $Omh^2$ value (method (I)) and $H_0 = 69.37\pm1.59$ ${\mathrm{km \
s^{-1} \ Mpc^{-1}}}$ that the $Omh^2$ value originates from Planck measurement
(method (II)), both at $1\sigma$ CL. Finally, we explore how the error in OHD
propagate into $w(z)$ at certain redshift during the reconstruction of $w(z)$.
We argue that the current precision on OHD is not sufficient small to ensure
the reconstruction of $w(z)$ in an acceptable error range, especially at the
low redshift
