Answer: 83

Solution:

Since $24 = 2^3 \cdot 3,$ a square $N^2$ is a multiple of $24$ exactly when $N$ is a multiple of $12:$ the factor $2^3$ forces $N$ to contain $2^2,$ the factor $3$ forces $N$ to contain $3,$ and conversely $(12m)^2 = 144m^2$ is always a multiple of $24.$ Also $N^2 \lt 10^6$ exactly when $N \lt 1000.$

The multiples of $12$ less than $1000$ are $12, 24, \ldots, 996,$ and there are $\frac{996}{12} = 83$ of them.