No you don't! The trail head starts a bit further away than what Google tells you but it is not trespassing

You can't really trespass in Iceland although there are some special caveats. If you're in a special place where, say, some special plant is being protected then you can be asked to move but Iceland is like most Nordic countries and you have a right to roam freely, even if the land is private. But you can't take the piss and set up a tent in someone's garden or annoy people, for instance. Nobody will shout at you unless you're doing something like fishing where you need a license or damaging wildlife.