Can time-invariance be determined from a given a transfer function?
The question can only be answered if it is clarified what it is that the function $G(z)$ describes. If $G(z)$ is the system's transfer function then we're done immediately, because only linear time-invariant (LTI) systems can be characterized by a transfer function of that form.
If $G(z)$ is the transfer function, the output sequence $y[n]$ is given by the convolution of the input sequence $x[n]$ and the impulse response $g[n]$, which is the inverse $\mathcal{Z}$-transform of $G(z)$. Note that in order to compute the inverse $\mathcal{Z}$-transform you must know the region of convergence (ROC), or you must be given the additional information that the system is causal.
Having computed the impulse response $g[n]$, the output is given by the following convolution sum:
$$y[n]=\sum_{k=-\infty}^{\infty}g[k]x[n-k]\tag{1}$$
It's trivial to show that the response to a shifted input $x[n-n_0]$ is given by $y[n-n_0]$. Hence, any system whose input-output relation is described by a convolution $(1)$ is time-invariant (and, obviously, also linear).