The snares exist in the game world from the time the quest is generated (or at least accepted), so you can find the snares while waiting for your traps to bag a bird.
This quest, like all the other "find things within area" quest, requires a tedious search through every tile within the area that matches the description: it does not have to be near the center.
I use two techniques for these area searches:
1. The preferred one is to sweep back and forth over the area on the world map, zooming in to every tile of the right type, look forward, look back, zoom out, move to the next tile, repeat. This covers an area is less than 5 days (but it generally takes more than one).
2. When the target is in spruce infested forest visibility is too poor for the previous technique to work. In that case I follow the borderline on the zoomed in map for as long as that feature is relevant, zoom out, move to the next one, etc. This technique generally takes longer and is more prone to overlooking a tile.
Thus, when going on these quests I make sure to bring food for at least a week.