There is one conjugacy class of elements of order 2, and if $p$ is odd then there are two conjugacy classes of elements of order $p$. This goes back to Dickson's 1906 book on Linear Groups.

There is one conjugacy class of elements of order 2, and if $p$ is odd then there are two conjugacy classes of elements of order $p$. This goes back to Dickson's 1901 book on Linear Groups.

Added later: for order $p$ elements this can be seen as follows. All order-$p$ elements of $PSL(2,q)$ are conjugate to elements of any prescribed Sylow $p$-subgroup. One such Sylow $p$-subgroup $S$ of $PSL(2,q)$ consists of the upper-triangular matrices with $1$'s on the diagonal. Now consider the action of $PSL(2,q)$ on the projective line $\mathbb{P}^1(\mathbb{F}_q)$. Each nonidentity element of $S$ has a unique fixed point, namely $\infty$. Thus, any element of $PSL(2,q)$ which conjugates one nonidentity element of $S$ to another must fix $\infty$, and hence must be upper-triangular. Finally, one easily checks that $$ \left(\begin{matrix} a & b \\ 0 & a^{-1} \end{matrix}\right) \left(\begin{matrix} 1 & c \\ 0 & 1 \end{matrix}\right) \left(\begin{matrix} a & b \\ 0 & a^{-1} \end{matrix}\right)^{-1} = \left(\begin{matrix} 1 & a^2 c \\ 0 & 1 \end{matrix}\right). $$ It follows that the conjugacy classes of order-$p$ elements are in bijection with $\mathbb{F}_p^\times/(\mathbb{F}_p^\times)^2$, so there are two such classes if $p$ is odd and one if $p$ is even.

For order $2$ I don't know a proof from first principles that is as short as the one above; the quickest proof I know is the one given in the proof of Lemma A.3 of my paper with Bob Guralnick titled "Polynomials with PSL(2) monodromy".