No; "kio" is a pronoun, not an adjective/determiner, so it can't modify a following noun.
Many English determiners are also pronouns and vice versa, but not so in many other languages.
One example where they are separate is possession - "your book" but "this book is yours".
So your sentence would sound as reasonable in Esperanto as "Yours book" would in English - simply the wrong kind of word for that position.
And "Kiu lernejo?" wouldn't be correct in this context, either -- right kind of modifier, but wrong case.
It would be fine in a context that demanded nominative case, of course - such as "Ĉu via lernejo estas bona?" - "Kiu lernejo?"