Understanding the structural and functional network architectures of nervous systems has been a key focus of neuroscience. Many features of brain-wide neuronal networks have been identified in mammals using noninvasive methods that lack cellular information. Here, we performed whole-brain calcium imaging in zebrafish larvae to investigate the structural and genetic basis of functional connectivity (FC) at single-cell resolution. Neural activity from $\sim$50,000 neurons was recorded in head-restrained transgenic larvae expressing a pan-neuronal calcium sensor, while monitoring their tail movements. From the regional calcium dynamics, we computed mesoscopic FC, revealing a conserved functional architecture across individuals. Despite this consistency, each larva displayed a distinct FC signature, allowing individual identification across imaging sessions on consecutive days. To characterize the structure-function relationship of brain networks, we used over 4,000 reconstructed neurons to derive inter-regional structural connectivity (SC). A strong correlation between SC and FC was observed, with polysynaptic pathways explaining much of FC. We identified structural network modules that significantly constrained the shape of both spontaneous and visually evoked regional coactivation patterns across individuals. Mapping stimulus- and motor-correlated neurons revealed a gradual organization of visuomotor populations along the anteroposterior axis, predicted by a network diffusion gradient in both spontaneous FC and SC. Additionally, we used spatially resolved gene expression profiles to identify a subset of genes whose region-specific co-expression levels significantly predicted FC. Our findings demonstrate that key principles of brain network organization established in mammalian systems can be observed at cellular resolution in zebrafish, highlighting its value as a vertebrate model for studying brain networks across scales.