Find the conjugacy classes in $\displaystyle D_4$.
Is there an easier way to do this than cutting out a square and checking every possibility?
You know in general that every element of the center is in its own conjugacy class. When calculating orbits of group elements it is also useful to note you will not get anything new from anything in the centralizer of the element. Lastly, keep in mind that conjugacy classes is an equivalence relation, so hopefully those things can trim your work down a bit.
Let $\displaystyle \rho$ be an element of order 4 in $\displaystyle D_4$ (i.e. a $\displaystyle \tfrac\pi2$ rotation).
Then $\displaystyle Z(D_4)=\{1,\rho^2\}$ so two of the conjugacy classes are $\displaystyle \{1\},\ \{\rho^2\}.$
Now $\displaystyle \rho$ and $\displaystyle \rho^3$ are the only elements of $\displaystyle D_4$ of order 4. Since elements in the same conjugacy class have the same order, and since $\displaystyle \rho$ and $\displaystyle \rho^3$ don’t belong to the centre (and so don’t form their own singleton conjugacy classes), $\displaystyle \{\rho,\rho^3\}$ must be another conjugacy class.
This leaves you with just 4 elements (the reflections) to check.