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 <<<

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Mathematics is a precise language. If you are vague, you'll get different answers.