Some armed forces administer general cognitive tests (like the US does with the ASVAB --- of course, not a random sample in the US, since we don't conscript) but most of these are not in fact IQ tests. Additionally, some western countries have done cross-sectional IQ tests for scientific reasons. In most countries, neither applies. Richard Lynn, responsible for the most widely known and cited "national IQ score" numbers, really did rely on mental health hospitals, and really did impute made-up scores to countries where he found literally no data at all.

What you do see are attempts to synthesize IQ from aggregate economic and educational attainment data. But obviously these are really just proxies for economic development, which then begs the question.