Choosing we get that the only values for which we get an integer matrix are , and the matrices are:
From the example above we can see that for any there is only a finite number of integer orthogonal matrices since their entries must be all either (why?).
The final observation for you to calculate the number of such matrices is that an integer orthogonal matrix must have only one entry equal to and all the rest equal
to in each column and each row (why? Check what'd happen with that column/row's length if there were two or more non-zero entries...)