Technically speaking, yes.
Practically speaking, impossible to find out
>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. Starting in Pokémon Omega Ruby and Alpha Sapphire, this also applies to gift Pokémon in the Undiscovered Egg Group.
Think of it in this way:
There are a total of 25 natures,
and a 50% chance of one nature.
This implies, the other 24 natures have a (50% / 24)
= 2.08 %
Now, assuming that the Modest nature was acquired without the boost, the chance would be (50% / 25) = an even 2%
So, the difference is a tiny 0.08%, which is practically impossible to distinguish from, because, it may be counted separately, or the chance was just "added over" over to the other 50%. Now, the question lies: Can a Pokemon be guaranted to have a nature more than 50% of time ? If yes, then it is simple. Chance of Modest nature is 50.08% meaning that the Ability did not activate. If not, tough luck, we still can't figure it out.
Here's to more confusion:
>In Generations III and IV, Nature is determined by the Pokémon's personality value. In later games, it is stored in an unrelated variable.
--Again, the pot-pourri of all Pokemon information.
>From Generation V onward, Nature is determined by a separate byte, unrelated to the personality value.
--Take a guess.
About Personality Values:
>A Pokémon's personality value is an unsigned 32-bit integer that is created when the Pokémon is first encountered. It is set when a Pokémon appears in the wild, when an Egg is first received from the Pokémon Day Care by the player (in Generation III* except in Pokémon Emerald and in Generation V) or by the Day-Care man (in Emerald and Generation IV), or when a Pokémon or its Egg is received from an NPC.
As an unsigned 32-bit integer, its value can be anywhere from 0 (32 zeroes in binary) to 4,294,967,295 (32 ones in binary), (both) inclusive.
Yay, more probabilities! (Attempt at sarcasm, for those of you who are sitting in the back)
So yeah, tough question, mate. You pulled off one of the most impossible to answer question ever. Congrats. Go celebrate with cake. (Not sarcasm. Cake is good.)
Finally, the answer to this mysterious byte that holds the answer:
It is formed by a pseudorandom number generator (basically, something so complex, that they gave it a name so long that I had to copy + paste that instead of actually typing it. Wowzah.).
More about this piece of extreme level algorithms that only Alan Turing and other such evolved minds can think of:
>In order to generate 'random' events in games and other forms of software, they must get as close to looking it as they can. There are limitless ways of accomplishing this. The degree of apparent randomness depends on the ability to predict the next result of the algorithm.
>First, the generator must have a seed, a number to start with. This number is usually a date and time referring to the first time that the algorithm is called during the usage of the device or the software's active session. Seeds are also occasionally derived from user input, as it is highly improbable to do the exact same thing more than once, making it appear 'random'.
This number is put through a complex algorithm, and the result is formatted according to the needed context. The raw result then becomes the seed for any subsequent uses of the random generator. Therefore, the nature of the generator is a recursive algorithm.
--Yea, you guessed it. 10 points to Griffindor. Yay.
Finally! Some answer! It all depends at what time you catch/ encounter your Pokemon! (lol.)