The relationship between genes and environment (or nature vs nurture) is highly interactive. Although other elements such as nutrition also plays a part, without stimulation from the environment, the child's brain growth or intelligence will be compromised and stunted.

This short phrase about brain development "use it or lose it" has strong merits, even for older people and Alzheimer's sufferers.

The logical conclusion is, a change in the environment or AN IMPROVEMENT in the interactive process can highly enhance a child's opportunities for better intelligence development.

The child's brain receives, stores and processes information signals, and produces outcomes for the practical purposes of survival and progress.

A healthy brain (the biological organ that they are born with) grows new networks as a result of information signals and stimulation from the environment.

The information and stimulation gathered by the 5 senses facilitates brain growth and intelligence development in children. The quality of stimulation and the child's ability to interact with that influences the quality of the child's brain development. 

So the short answer is both genes and environment play critical role in a child's intelligence development and modifying the environment can impact or enhance this development