No you cannot, not without cheating/hacking anyhow.
Chat with our AI personalities
Go to the house next to the Poke Mart in Eterna City. Talk to the old man and he will change your pokemon's nickname. Sadly, you cannot change the name of trade pokemon.
You can go to the name rater in Lavender.
Unfortunately, you cannot nickname traded Pokemon in this game, or any Pokemon game for that matter. You can nickname your own caught Pokemon by going to the Condominiums next to the PokeMart in Eterna City and ask the old man near the entrance. I hope that you will make do!
Go to Eterna City and go into the building, that has the Name Rater on the sign. Talk to the old dude at the table and he will "rate" your pokemon's nickname and ask if you want to change it. You cannot change a traded pokemon's nickname.
In the large building in Eterna City on the first floor. He is the old man. Remember, you can't change the nickname of a traded Pokemon.