From the Wikipedia page that you linked:
"There has been significant controversy in the academic community about the heritability of IQ since research on the issue began in the late nineteenth century"
"explaining the similarity in IQ of closely related persons requires careful study because environmental factors may be correlated with genetic factors."
"The heritability of IQ increases with the child's age and reaches a plateau at 18–20 years old, continuing at that level well into adulthood. However, poor prenatal environment, malnutrition and disease are known to have lifelong deleterious effects"
"Although IQ differences between individuals have been shown to have a large hereditary component, it does not follow that disparities in IQ between groups have a genetic basis.[10][11][12][13] The scientific consensus is that genetics does not explain average differences in IQ test performance between racial groups"
And it continues in a similar way, so there is some genetic basis, but at the same time epigenetic factors seem to matter a lot.
The issue is the misuse of the term "heritable" to mean "genetically determined". Toe count is genetically determined; virtually all people are coded for 10 toes. But they have very low heritability: variance in toes is environmental. Conversely: dress-wearing isn't at all genetically determined; anyone can put a dress on. But dress-wearing is highly heritable: the variance is (almost) entirely due to genetic differences.
