"in front of the [tex]y,z[/tex] plane" probably means [tex]x\ge0[/tex], in which case
[tex]4x^2-4y^2-z^2=4\implies x=\sqrt{1+y^2+\dfrac{z^2}4}[/tex]
We can then parameterize the surface by setting [tex]y(u,v)=u[/tex] and [tex]z(u,v)=v[/tex], so that [tex]x=\sqrt{1+u^2+\dfrac{v^2}4}[/tex].
