Is there a source for the “segula” to have a child by being kvater and kvaterin at a bris?
Not that I know of. It is generally believed to be a Segulah, but the source for this is unclear.
The idea of the Kvater is that really the mother should pass the baby to her husband (who serves as Sandak), thus playing her role in the Sandakaus (see Rema, Yoreh De’ah 265:11). However, the mother is generally assur to her husband at the time of the bris, and therefore she passes the baby to a married lady, who passes on to her husband, who passes on to the father.