I originally wanted to name Fry and Bender, but my wife wouldn't let me.
I managed to get my wife to let me name a chinese hamster Fry, after the Futurama character, couldn't get bender past her though. We also had a Roborovski hamster named "Wodka" as in "Chekov saying Vodka"
I had a goldfish named Rimmer after the Red Dwarf character, and also tried to name our cat Jazz, after a Transformer but Mrs Baron said no....we settled on Willow...which could be the dwarf I guess.
But no, no trek pets - though I know someone who once had a Rat called Sela.