If he holds the domicile of Uttar Pradesh, then yes he's eligible for state quota. If he holds the domicile of some other state then he'd be eligible for the state quota of that particular state. And hence it doesn't actually matter from where has he given his class 12 board exams.

The domicile of UP can be obtained by you, if you are a permanent resident of Uttar Pradesh or have been residing there for atleast a span of three years.  You need to approach District Magistrate or any Additional. District Magistrate/ SDM authorised by the DM to issue Domicile Certificates.