I didn't used to back in the day, but nowadays if I think I'm going to raise it on my team, I give it a nickname. It usually takes me a little while to come up with them, though, and often I'll use real names, such as through the website Behind the Name, or try to put something together using the letters of the final evolution's species name. Sometimes it just comes to me the second I look at the Pokemon though, such as the Treecko I raised for my first play through of Alpha Sapphire, whom I nicknamed Richard simply because he looked like a Richard to me.
As far as the nicknames of my main 3DS team? Scarlet (Delphox), Abyss (Absol), Cairo (Lucario), Lycea (Lycanroc [Midnight]), Ysayle (Ninetales [Alolan]), and Emma (Noivern).