The question provides four statements about the 44th Chess Olympiad, 2022 and asks how many are correct.

1. The statement that it was the first time the Chess Olympiad was held in India is correct.

2. The statement that the official mascot was named Thambi is incorrect. `Thambi` is mentioned nowhere in the official olympiad records.

3. The statement that the trophy for the winning team in the open section is the Vera Menchik Cup is incorrect. The Vera Menchik Cup is actually given to the winner of the women`s section.

4. The statement that the trophy for the winning team in the women`s section is the Hamilton-Russell Cup is incorrect. In actuality, the Hamilton-Russell Cup is given to the winning team in the open section.

Therefore, out of all the given statement(s), only one is true. Hence, the correct answer is "1, Only one".