@fBoyle-- Are your nails yellow and flaky as well? If so, that does sound like a fungal infection and yes, fungi can cause pus. It's not as common as it is with bacteria, but definitely possible.
By the way, do you have any ingrown nails or any other nail problems? Ingrown nails can cause pus too because the nail can cut into the side of the finger and cause an infection.
You should see a doctor for a diagnosis. He or she will tell you what to do and what medications to use. If you try to treat it yourself, you might use the wrong thing and the infection will get worse.
I had a finger infection because of an ingrown nail a few months ago. My doctor cut the nail, drained the pus, cleaned the area and packed it for me. I wouldn't have been able to do that at home. I also have diabetes, so leaving it untreated would have been extra dangerous for me.