Quickest if definitely a taxi. That will run you around $15-$20. If you want quickest & free using Disney transportation, well, there's nothing "quick!"
I would take the bus from POFQ to DHS, then take the boat from DHS to BC. You do not need to enter DHS to do this, so no park hopper necessary. You could also walk to BC from DHS, it's a good 20-25 min.
You could take the bus from POFQ to Epcot, walk through Epcot, out the IG in WS, and straight on to BC. This will require a park hopper as you need to enter Epcot for this. This entails a bit of walking as well.
As a last resort, you could take the bus from POFQ to DTD, then transfer there to the BC bus. I have always had terrible wait times with any DTD buses, so this could take awhile.