Walter asked for a cow in the pilot episode, "Pilot". It's unknown who named it Gene.
According to the transcript at Fringepedia:
Harvard Lab - Setting Up
OLIVIA: Dr. Bishop, I've ordered a standard forensics' work package. Is there anything else you need?
WALTER: Optical coherence tomograph for flesh study. Two thousand pounds of silicon. At least five anonymous blood samples from volunteer donors, micro-organism detector. NASA grade, of course. There's still NASA, yes?
WALTER: Oh, she's still here. This tank was the best. And a two-Year-old bos taurus.
ASTRID: A what?
PETER: A cow. He wants a cow.
WALTER: Purebred, not a crossbred, this is important. Mature weight 850 pounds, total fat average 2.37.
OLIVIA: Is he joking?
PETER: Genetically, humans and cows are separated by only a couple lines of DNA. So, what's an ethical test subject?
OLIVIA: Where'd you learn that? MIT?
PETER: No actually, I picked that up reading books. You should try it sometime. It's fun.
OLIVIA: (to Astrid) Get him the cow.
WALTER: Fantastic, thank you. Only thing better than a cow is a human. Unless you need milk. Then you really need a cow.
(Activity including Walter examining laboratory slides; Olivia reviewing photos of Flight victims; an unidentified, and unrelated, class in session; and the arrival of the cow.)
Harvard Lab - John Scott Arrives
CHARLIE: Is that a cow?
OLIVIA: Yeah, that's Gene.