What are the reduced elements of $\mathbb Q(\sqrt{30})$? 
What are the reduced elements of $\mathbb Q(\sqrt{30})$ ?

From the definition here(on page $32$);
An element $\beta\in\mathbb Q(\sqrt{d})$ is said to be reduced, if $\beta>1$ and $-1<\beta'<0$ and $\beta'$ is the image of $\beta$ under the nontrivial automorphism which sends $\sqrt{d}$ to $-\sqrt{d}$ 
If $\beta$ is of the form $a+b\sqrt{30}$, then $0>\beta'=a-b\sqrt{30}>-1$ 
this looks too complicated to determine all reduced elements
So if $\beta$ is reduced, then clearly it is irrational, and we've a theorem which states: For $\beta\in\mathbb Q(\sqrt{d})\setminus\mathbb Q$, there are only finitely many reduced elements for any $D\in\mathbb N_{>0}$, where $D$ is the discriminant $(B^2-4AC)$ of the minimal polynomial $(AT^2+BT+C=0$ and $ \gcd(A,B,C)=1)$ of $\beta$. Is this useful ?
For example if we define $\beta_0=\sqrt{30}+\lfloor\sqrt{30}\rfloor,\quad\beta_{n+1}=\frac{1}{\beta_n-\lfloor\beta_n\rfloor}$ than $(\beta_n)_n$ is a periodic sequence of reduced elements of $\mathbb Q(\sqrt{30})$, but these are not all.
Do you have an idea ?

If I understand the definition correctly, all you need to do is solve the system of inequalities $$a+b\sqrt{30}>1\text{   and   } 0>a-b\sqrt{30}>-1$$ for rational numbers. You can't of course expect to list the solutions -there are infinitely many. It is not difficult to describe the set though:
The first inequality is a half plane; to see that notice that you can rewrite is as $(a,b)\cdot (1,\sqrt{30})>1$ which means the projection of $(a,b)$ on $(1,\sqrt{30})$ is more than $\frac{1}{\sqrt{31}}$. This is a half-plane, with boundary orthogonal to the vector $(1,\sqrt{30})$ cutting it at $(\frac{1}{\sqrt{31}},\sqrt{\frac{30}{31}})$.
The second inequality is similarly an infinite strip vertical to the vector $(1,-\sqrt{30})$. The picture would look like this:

Now, to give a maybe more precise answer you can definitely parametrize this domain. Notice that the lines $a-b\sqrt{30}=-1$ and $a+b\sqrt{30}=1$ intersect at point $(0,\frac{1}{\sqrt{30}})$ and the other two intersect at $(\frac{1}{2},\frac{1}{2\sqrt{30}}$).
The domain is then parametrized as the union of sets $A$ and $B$ where $$A=\big\{(a,b):a\in[0,1/2]\text{ and }b\in[\frac{1-a}{\sqrt{30}},\frac{1+a}{\sqrt{30}}]\big\}\text{ and }$$
$$B=\big\{(a,b):a\in[1/2,\infty)\text{ and }b\in[\frac{a}{\sqrt{30}},\frac{a+1}{\sqrt{30}}]\big\}$$
Of course, you want the points with rational coordinates there...
