The only correct option is children's education. 'Children education' is wrong.

Actually both are correct and the meaning depends on what you want to say, if you say children’s education you’re talking about the education of children for example your children or the children you know or whatever, but if you say children education you’re talking about the education which is about children, not specifically the children you might know but generally talking children

Abdullah is right: 'children's education' is correct and 'children education' is wrong.

I'm afraid that the first answer you received was wrong, and we're still waiting for Victor to come back to us with the examples he found. I suspect that his examples were one of two things:

- inaccurate non-native English. The internet has thousands of sites with content written by non-native English speakers which look professional but are full of mistakes.

- sentences where the two words happen to come next to each other, but are unrelated.

'Children education' is wrong because it is incorrect grammatically. Here's why:

When we make a compound noun composed of two nouns, the first is singular and has no apostrophe . This is why we can say 'science education', but not <s>sciences education</s> or <s>science's education</s>. We can say 'college education', 'adult education' or even 'child education', but you can't say 'children education'. 'Children' is a plural noun, and you cannot make a compound noun where the first (and more specific) noun is plural. The rules of English noun formation do not allow this.

Think of some simpler examples: we say 'shoe store' and 'cherry pie' even though there are lots of shoes in the store and lots of cherries in the pie, don't we? This is because of the way we have to form compound nouns. It is incorrect to say <s>shoes store </s>and <s>cherries pie</s>, and it is incorrect to say 'children education' for the same reason.

If you want to use the plural noun, we have to make a noun phrase where the first noun is in the genitive case, with an apostrophe 's': students' education, women's education, children's education. These are grammatically different from the examples above, and you cannot omit the 's' or the apostrophe.

As it happens, I've just answered a question on the same point here: https://www.italki.com/discussion/212951, if you'd like to see some more similar examples.

I hope that all makes sense now.