It can! Please think twice before ordering osmium. The oxide is volatile and extremely toxic: one good huff is enough to give you severe pulmonary edema, and make you drown in your own body fluids several hours after the exposure. Contact with eyes will make you permanently blind by turning your corneas black and opaque.
However, the most danger is in handling osmium powder. The real danger is in handling osmium powder. Meanwhile, the question explicitly suggests that the bulk form of osmium is being considered: OP says "I love shiny rocks", not "I love dull gray powders". Also, all the sellers of elements for collection that I know of tend to sell osmium in the bulk form: cubes, pellets and nuggets, not in powdered form. I wish more people were actually reading the questions, instead of just quickly glancing over the title and copy-pasting two sentences from the first Google hit.
Formation of $\ce{OsO4}$ from osmium metal in room temperature is thermodynamically unfavorable. Handling bulk osmium is not nearly as dangerous. Be careful, though! While the majority of metals is ductile and malleable, some exceptions -- including osmium -- are brittle. Instead of deforming under mechanical stress, they shatter into small pieces, greatly increasing the surface area that could react with oxygen in the air!
Here is a video of a man handling bulk osmium nugget. Notice that he is wearing gloves. He has since been recently active, and is alive and well.
Wikipedia says:
$\ce{OsO4}$ is formed slowly when osmium powder reacts with $\ce{O2}$ at ambient temperature. Reaction of bulk solid requires heating to 400 °C.
It is a bit of a shame that the accepted answer does not really make any effort to actually answer your question and does not even differentiate between powdered and bulk osmium. Instead, it just copies two short sentences from an external source.