I agree with Zoe. I thought I'd follow-up to explain the reason why, "Put the gear in neutral," is correct.
When you're shifting gears, as you talk about in your sentence, you're putting the gear of the car in a position: park, reverse, neutral, drive, low, etc. As you're aware, but what is good to point out as it relates to the English language, is that these "positions" are arbitrary. They are words (e.g., "Neutral") used to communicate to the driver what is happening mechanically in the car. So, what is implied in any of these cases is the word position. See below:
"Put the gear in (the) park (position)."
"Put the gear in (the) reverse (position)."
"Put the gear in (the) neutral (position)."
"Put the gear in (the) drive (position)."
"Put the gear in (the) low (position)."
However, we are so accustomed to using these words (park, reverse, neutral, drive, low, etc.), that we do not need to say "the ____ position" to communicate to someone else. Instead, we can use "neutral," like you did in your sentence, to communicate. This is grammatically correct, too. No problems there.
If you wrote it the second way ("Put the gear on neutral"), the gear now becomes a physical entity that you are placing on "neutral." We can both agree that this is not the case, and thus, it is an incorrect way to write the sentence. We are not picking up and placing the gear on "neutral"; rather, we are using the gear shift to change how the car operates (quite frankly, the actual mechanics is beyond my understanding, but I think you see the point).
Lastly, here's another way to think about it. When we say "put the gear" we are really using "put" as a synonym for "shift." As mentioned, we are shifting the gears of the car with the gear shift. The sentence wouldn't make sense if it was worded this way: "Shift the gear on neutral." If you have a question regarding this sentence, please feel free to comment and I can explain more.
I hope this helps!