Does GDP include education?
Yes: GDP is the summed added value of all produced goods and services, this includes education.
Is the US the largest education exporter in the world?
The US are a big country, so you might want to correct for scale. Here is a graph for public spending on education, as a share of total GDP
As you can see, I randomly added Germany, Sweden and the World average. The US is clearly not the countries with the highest public spending on education, as a share of total GDP.
However, if you want to look at education exporter, you need to look at something different. There should probably be statistics on the number of foreign students in US universities. Comparing the number of foreign students in each country (and assuming that the share of those that go back home is constant between countries) should give you an answer to the question.