Infant Island is the home of the miniature Shobijin Priestesses and the guardian of Infant Island, Mothra, whom they serve. The first appearance of the island is in the original Mothra film from 1961.
Infant Island appears to be an island in the South Pacific and is only inhabited by the cult that worships Mothra.
Strangely, in the film Mothra vs. Godzilla it has been devastated due to being used as a site for nuclear testing after the second world war; the atomic energy unleashed has reduced it to a nightmarish landscape of weirdly warped rocks, pools of toxic-looking water, and bleached skeletons. While this is not necessarily incompatible with its portrayal in the original Mothra film, it is portrayed as still being a lush tropical paradise in subsequent Godzilla films of the Showa era, suggesting that the damage incurred by the island was retconned out of the Showa continuity.
In Godzilla vs. Mothra, Infant Island is located in South-East Asia, specifically the Philippines (or possibly Indonesia).