When you enter Wintergrasp, you will be PvP Flagged. (It will remain for 5 minutes after you leave Wintergrasp.)
Wintergrasp is a contested zone, even where the factions are normally at peace (PvE realms). Fishing in Wintergrasp is indeed dangerous! Fish in-between battles to avoid most enemies. Battles occur about every 3 hours and last up to 30 minutes. The timer can (with patch 3.1.0) be seen on the map.
Your faction does not need to control Wintergrasp to complete this quest.