To be precise: none, 192, or an infinite number.
If you had stated:
1. How many zero integers are there, between 1 through 1,000, the answer is NONE.
Note: BY DEFAULT, all math questions assume Real unless you specify otherwise.
2. If you included all REAL and/or Complex '0' digits, the answer is INFINITE.
3. If instead, you asked (and I think this is what you meant):
"How many zero digits are included in the integer numbers between 1 through 1,000"
The answer is 192 <<<
Mathematics is a precise language. If you are vague, you'll get different answers.