Answer
No, flux through the Gaussian surface does not change.
Work Step by Step
By Gauss law, flux inside a Gaussian surface enclosing charge $q_{enc}$ is given by:
$$\phi = \frac{q_{enc}}{\epsilon_{0}}$$
where $\phi$ is the flux.
Clearly, flux $\phi$ does not depend on the location of charge inside the body.
Hence, as long as the charge is inside the Gaussian surface flux remains the same.
