I'm at the island but no one is telling me to visit the captain and thus start the quest. I've used every dialogue option available for basks in sun and the guards, and even talking to the captain failed to start the quest.
Am I doing something wrong?

I just played through it for
I just played through it for the first time without any difficulties. If you have spoken to Carius, and did not get the dialogue to ask about low morale, I would like you to check if the quest has stated by using console commands. Press ~ and type "getstage fbmwbmmorale" and press enter. If it comes back 0, it hasn't. I am relatively new to Oblivion/Morroblivion, but am experienced with Skyrim. I know many people do not like setstage, but it has worked for me dozens of times without breaking my game. A possible workaround could be to enter the command "setstage fbmwbmmorale 10", and try speaking with Carius again. If nothing is different, try, "setstage fbmwbmmorale 20", which will put you to the part where you have to give a soldier a drink. If you get stuck, consult
http://www.uesp.net/wiki/Bloodmoon:Rebellion_at_Frostmoth