We've had a similar question to this before, but it was done injustice.
Anyways, the answer is no it will not work on Gift Pokemon.
You have stated that it only works on gift Pokemon, which is not an encounter in the wild.
"From Pokémon Emerald onwards, if a Pokémon with this Ability is leading the party, there is a 50% chance that wild Pokémon encountered will have the same Nature as the Pokémon with Synchronize. This does not apply to roaming or gift Pokémon. In Emerald, it did not apply to stationary Pokémon either, such as Regirock or Rayquaza." - Bulbapedia