The relationship between a child's genes and their environment (or nature vs nurture) is highly interactive. Both are needed and both affect a child's intelligence development. 

Although other elements such as nutrition also plays a part, but without stimulation from the environment, the child's brain growth or intelligence will be compromised and stunted.

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

The interaction of the 2 factors suggests that a change in the environment or stimulation, or AN IMPROVEMENT in the interactive process can highly enhance a child's ability to develop higher intelligence

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 stimulation FEEDS the brain. The quality of stimulation, therefore, and the child's ability to interact with that stimulus influences the quality of the child's brain development. 

Both genes and the environment play critical role in a child's intelligence development and modifying the environment can modify and enhance this development