When you meet Faralda at the bridge to the College of Winterhold, there is an option to persuade her to let you across the bridge. However, if you succeed, the misc. quest "Visit The College of Winterhold" will not complete and will never be completeable. Is there a way to fix this, with a console command/etc? I'm playing on PC.
use the console command
setstage MG01Pointer 200
to complete this quest.
There is actually a mod that I feel may be better and worth a shot for fixing this called the 'Unofficial Skyrim Patch' .. and you can find a version for all the DLCs so far. This collection of mods is just a set of community patched bugs and one of the things they are very good at clearing up are hanging quests like this as well as potential progression breakers and the like.
The mods can be found both at Skyrim Nexus and the Steam Workshop.
Hope this helps.