The standard connection size for regulators, etc. is 1/4" NPT (National Pipe Thread). There are some larger connection sizes, but the 1/4" NPT is the most common for smaller compressors.
I don't know exactly how the hose is connected to the air tank, but what you need to find is an adapter that goes from whataver that connection is to 1/4" NPT and you should be good to go. It is possible that the hoses are threded onto a nipple on the tank and that nipple will already be threaded properly.
Most larger hardware stores, Lowes, Home Depot, etc. have a good selection of pipe sections, adapters, etc. Keep in mind that it does not have to be specifically for a compressor, any hunk of pipe will work as long as it's threaded properly so don't just look in the compressor area.
Depending on your airbrush hose you may need another adapter to get to 1/4 NPT. My braided Badger hose is already at that thread size, but my vinyl hose isn't. Badger makes an adapter that goes from 1/4 NPT to their little bitty hose size (whatever it is!).