Answer: 68

The smallest integer with 4 digits is 1000, 1 more than 999 which only has 3 digits.

The largest integer with 4 digits is 9999, 1 less than 10 000 which has 5 digits.

10 000 = 100$^2$, so all integers whose squares have 4 digits must be less than 100.

30$^2$ = 900, 31$^2$ = 961 and 32$^2$ = 1024.

So all integers $n$ with 31$\lt n\le$99 have 4-digit squares.

So there are 99 $-$ 31 = 68 such numbers.