I started working in maintenance at an apartment and I was concerned with high dust exposure, could there be asbestos in the walls?

Pre-1989 or after? How ancient is the apartment building? Asbestos was banned in the U.S. In 1989. Hard to tell asbestos by just looking. Many counties and states have inspectors if you're worried. Since it's uber-dusty, how about wearing some breathing protection? You could try a dust mask labeled n95 or a niosh-approved respirator.