Dear student,
Yes, during the S-phase of duplication, the duplication of the centrosome as well as of the centrioles takes place.
It takes place in the following way :
1. Centriole disengagement- In this, the tight link between the mother and the daughter centriole is broken. Though, they remain connected through loose fibrous structure.
2. Centrosome duplication- Here the the mother and the daughter centrioles duplicate themselves. New centrioles are formed at the proximal ends. This occurs during the S-phase.
3. Centriole engagement- The full length of the newly formed centrioles is reached. The old daughter centriole becomes the new mother centriole and the link is broken.
4. Centrosome maturation- The two centrioles that makes up the centrosome now pairs and collect other materials for further maturation.
5. Centrosome separation occurs.
Thus, centrioles become 4 in number and centrosomes become 2 in number. after completion of S-phase.
Regards.