I turned over a good compost pile with my tractor yesterday. It's about a big pickup truck load of grass clippings, oak sawdust, and chopped-up oak leaves from last fall and it's set all winter. It's on the way to being real good compost.
In turning it, there were some layers of grass clippings that hadn't rotted down - they probably weren't getting air before I turned the pile. The grass put little clouds of gray dust in the air when disturbed. The pile isn't hot so it wasn't smoke, and those areas of grass clippings are gray also.
What is that stuff? I've seen it before, and I'd guess it's mold or mildew spores - probably not a good thing to breathe.
Gray dust from compost
Fungal spores. And nope, avoid inhaling them. Pathogens such as legionella love this environment.