Yup you can cure it from a quest, and I think the cure might he permanent... You caught it from an enemy in a fight, there are a number of diseases you can pick up while fighting enemies. The game will flash you a status message that you contracted x disease, but the messages are easy to miss. If you click Magic > Active effects menu, it will show you any disease you have in red. Townspeople will also make comments like are you feeling unwell or you look ill.
You can cure any disease you contract with a Cure Disease potion (carry a few with you), or go to the temple in any major city and activate the shrine and heal for free. It's a good idea after every quest to hit a temple when you go to town to sell loot... Just in case. In Whiterun the temple is right above the shops, up the steps and behind the old tree. The shrine is in the back and looks like a pastel bike helmet. Click it to activate and you'll see the healing blessing take effect.
You usually have a couple days of game time until diseases will get full blown. Now you have to do the quest to get rid of the vampire bug, nothing else will cure it.