If the flux of the electric field through a closed surface is zero, then

By Gauss's law, electric flux through any closed surface is given by,

$\mathrm{\varphi }=\text{∮}\overrightarrow{\mathrm{E}}\cdot \overrightarrow{\mathrm{dA}}=\frac{{\mathrm{q}}_{\text{enclosed }}}{{\mathrm{ϵ}}_0}$

  $\mathrm{\varphi } = 0$$\Rightarrow {\mathrm{q}}_{\text{enclosed }}=0$

As the flux is zero through the surface, the charge enclosed must be zero. But the electric field is not necessarily zero everywhere on the surface. For example, in the case of a dipole enclosed by the surface, the electric field through the surface is not zero .