Question

When a star enters the red giant stage what does it begins to convert in its core?

Answer 1

In a red giant's core, nuclear fusion will turn helium into carbon.

Explanation:

After the star's core runs out of hydrogene, it will no longer produce radiation to balance the star's weight.
The star will collapse, the core will contract and its temperature will rise.
If the core's temperature rises high enough, nuclear fusion will create carbon out of helium in what is called "the triple-alpha process":

two helium nuclei will fuse to create an unstable berylium nucleus, that will fuse with a helium nucleus to create a stable carbon nucleus.