Depends how strictly you stick to definitions. Geordies are from Tyneside, so as Hexham is by the Tyne, then theoretically, this could be said to be true.
However, most people from the North East accept that Geordies are "Novocastrians" ie, born in Newcastle.
(FYI, South Shields is at the mouth of the Tyne, but people from here are known as Sanddancers)
HTH,
A geordie is a person born in Necastle, on one side of the river (sorry i googled and searched this site [this question has been asked before i am sure] but can't find whether it's south or north side but my money is on north side). The other side of the river you are not traditionally allowed to be known as a Geordie! Anyway these days the term Geordie refers generally to people of Newcastle (whichever side of the river) BUT NOT Sunderland - they're called Mackems! Therefore in answer to your question no you're not really a geordie if born in Hexham. ps. Geordie comes from the nickname given to the George Lamps used down the mines