Can I wall mount this pedestal sink?
Hung sinks typically have brackets that connect high up on the sink. (This makes sense, because that's where you'd want to resist leverage.)

If you could somehow fashion a 2 piece bracket, it's likely** (see below) that you could drill into the porcelain (without breaking it), and attach it using sturdy toggles.

Unfortunately, it looks like you have a wall that's already tiled, so you don't have any depth into which to set this bracket. So unless you want to bust up your wall, I'd suggest you stick with the pedestal. Or get a deliberately designed sink that'll have the recess for the bracket built into the sink.

Porcelain is an inconsistent product, so failure is always an option. But, a diamond hole saw, used according to instructions (speed, spraying with water while drilling) will usually drill a successful hole. Plus, in this setting, a major blowout would probably be limited to the invisible back surface, which would mean that the sink is back to being used as a pedestal, which doesn't seem like such a great loss to me.