An animal can't produce gluten, right?

This might be a crazy question, but it's been bothering me. A local farmer mentioned that his most sensitive gluten-free customers ask for a specific variety of turkey that doesn't eat gluten-containing grains so it's safer for them than turkeys that eat gluten. I'm no biologist, but I cannot imagine a way an animal cell could contain gluten - though I suppose if there was contact between the meat and stomach contents during butchering that could be an issue? Is this someone anyone has heard before? My state is not known for our scientific literacy, so I've chalked this up to that, but ever since hearing this my family is now bring it up to me (the lone Celiac) as something I should consider and it's driving me mad.