Previous studies have shown that language acquisition influences both the structure and function of the brain. However, whether the acquisition of a second language at different periods of life alters functional network organization in different ways remains unclear. Here, functional magnetic resonance imaging data from 27 English-speaking monolingual controls and 52 Spanish-English bilingual individuals, including 22 early bilinguals who began learning a second language before the age of ten and 30 late bilinguals who started learning a second language at age fourteen or later, were collected from the OpenNeuro database. Topological metrics of resting-state functional networks, including small-world attributes, network efficiency, and rich- and diverse-club regions, that characterize functional integration and segregation of the networks were computed via a graph theoretical approach. The results showed obvious increases in network efficiency in early bilinguals and late bilinguals relative to the monolingual controls; for example, the global efficiency of late bilinguals and early bilinguals was improved relative to that of monolingual controls, and the local efficiency of early bilinguals occupied an intermediate position between that of late bilinguals and monolingual controls. Obvious increases in rich-club and diverse-club functional connectivity were observed in the bilinguals relative to the monolingual controls. Three network metrics were positively correlated with Spanish proficiency test scores. These findings demonstrated that early and late acquisition of a second language had different impacts on the functional networks of the brain.