Are those spots on my plant at my desk at work anything to worry about?
Your plant appears to be Zamioculcus zamiifolia (more easily known as ZZ plant). These dark blotches or spots are quite normal and nothing to worry about - they develop on older stems. The stem I'm uncertain about is the more yellowish coloured stem with blotches in the forefront of the picture, that one might have a problem, because of its yellowish colour rather than the blotches, although the leaves coming off it look perfectly healthy. If the leaves start to turn yellow, then clip that stem out and adjust your watering as described below, if that's not how you're watering currently.
Getting the watering right is important - they need to dry out a little between waterings, so water when the surface of the potting soil feels just dry to the touch, water well, and empty out any excess from an outer pot or tray 30 minutes later. They tolerate various light conditions, but prefer relatively bright light, though never direct sunlight, and temperatures between 13-26 deg C. Usually, overwatering shows itself by yellowing leaves rather than a yellowing stem; the colour of that stem may not be particularly significant, though the need for repotting might be a possibility, I'm unable to tell because the pot is not visible in the photo. More information here https://www.ourhouseplants.com/plants/zzplant
The whitish spots look like pesticide residue to me. This is most likely on a new plant. One of the last things the growers do before shipping is an application of pesticide to take of insect pests. It could also be dissolved salts from alkaline water.
Take a cloth and rub the leaves, if it comes off it is a residue. Wash your hands afterwards in case they used something with residual activity.
Otherwise your plant looks very healthy