Z domain transfer function to difference equation
If $x[n]$ is the input of your discrete-time system and $y[n]$ is the output, then the transfer fucntion H(z) is written as: 
$$H(z)= \frac{Y(z)}{X(z)}$$ 
where $$X(z)=Z(x[n])\; \;\; \;,\; \;\; \;\; \;Y(z)=Z(y[n])$$
So we get: $$\frac{Y(z)}{X(z)} = \frac{2(z-0.5)(z-0.6)}{z-1}\Rightarrow (z-1)Y(z)=2(z-0.5)(z-0.6)X(z)$$
$$\Rightarrow zY(z)-Y(z)=2z^2X(z)-2,2zX(z)+0,6X(z) $$
At this point we apply Inverse Z Transform and make use of both Linearity and Time Shifting properties. 
So we end up with the following difference equation:$$\mathbf{y[n+1]-y[n]=2 \cdot x[n+2]-2.2 \cdot x[n+1]+0.6 \cdot x[n]}$$