Output impedance refers to open circuit o/p impedance or short circuit o/p impedance?
The variable g is used for what are called the inverse hybrid network parameters. It is indeed defined as 

\$g_{22}=\dfrac{v_2}{i_2}|_{v_1=0}\$

The inverse hybrid parameters are very rarely used in practice, and they don't have conventional names except "the inverse hybrid parameter \$g_{22}\$".

The z parameters are conventionally called the impedance parameters. If someone tells you they are giving you an impedance value, it should mean the z-parameter if they don't tell you otherwise.

In many useful networks (for example, well-designed amplifiers) there is sufficient directivity such that it doesn't really matter whether the input is open-circuited or short-circuited when measuring or specifying the output impedance. But for general networks you should indeed consider the input as open-circuit when determining \$z_{22}\$.


  Is thevenin equivalent impedance, seen from output port has any connection with g22 or z22 ? 


Let's look at the equivalent circuits implied by the two types of network parameters:







simulate this circuit &ndash; Schematic created using CircuitLab

You can see that the Z-parameters describe a network as if it has a Thevenin-like circuit at each port, with the source value being dependent on the current in to the other port.

For the inverse hybrid parameters, there's a Thevenin-like circuit on port 2, but a Norton-like circuit on port 1.

Looking at these equivalents also shows you why you have to take the controlling variable from the other port to be 0 to find \$z_{22}\$ or \$g_{22}\$. If you didn't take the other independent variable to 0, the output wouldn't depend just on what's connected to port 2, it would also depend on the conditions at port 1.