Converting a number to a member of a multiplicative cyclic group
Use the Legendre symbol in your case.
Explanation: when $p=2.q+1$, the order of the multiplicative group of $F_p$ is p-1=2q.
Then there is no other alternative, when you select a random number, it could be a quadratic Residue with probability $\frac{1}{2}$.
The legendre symbol of any number a, is : $a^{\frac{p-1}{2}}=a^{q}=\pm 1$

Then except the number 1, all the other quadratic residue (Legendre symbol is 1) have order q.