No, you cannot simply replace where with a relative pronoun or ‘null relativizer’.
This is because where itself is not a relative pronoun—it does not ‘stand for’ a noun or NP—but a relative adverb: it stands for a locative, a preposition phrase. Where in your initial sentence does not represent ‘the place’ but ‘to the place’.
Consequently, when you replace where with a pronoun you must at the same restore the preposition which is ‘embedded’ in its sense:
You should try the barber’s that I go to OR
You should try the barber’s I go to OR
You should try the barber’s which I go to OR
You should try the barber’s to which I go.