After you talked to the doctor you should go up to his lab using the turbo-lift next to him, which has everything you need to complete the quarantine mission. There are several computer consoles in the lab that you can interact with. One of them lets you communicate with Memory Alpha to ask about a chemical spill accident that is related to your contaminated cargo. Workers on Memory Alpha accidentally found a cure because someone broke a chemical container. You should read all the reports carefully as they all contain clues as to how to make the cure for your crew.
After you read the reports you then go to another console to experiment with the variables. Your goal is to find a type of material that will contain the bacteria and a chemical that will erradicate the bacteria. If you are successful the computer will tell you so and you can then proceed to another console to create your cure. However, this is the tricky part, some of the chemicals listed in this console is deadly to humanoids, so you have to go with your gut instinct and hints from the reports.
Finally, you need to test your cure on the quartermaster who is quarantined behind a force field right inside the lab. There was a bug when i was there and the quartermaster didn't show up on my screen, but you could still administer the cure to the force field. Mission ends as soon as you revive the quartermaster. If the quartermaster writhes in agony then you've failed and you need to try a different combination of materials and chemicals.