I interpret the question "how does one find all the relations between the matrices" as "find a set of defining relations for the group generated by the two matrices".

To do that we need a presentation of $GL(2,\mathbb{Z})$. I found one in the paper:

T. Brady, Automatic structures on Aut$F_2)$, *Arch. Math.* 63, 97-102 (1994).

$\langle p,s,u \mid p^2=s^2=(sp)^4=(upsp)^2=(ups)^3=(us)^2=1 \rangle$

where $p= \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$, $s= \left(\begin{array}{cc}-1&0\\0&1\end{array}\right)$, $u= \left(\begin{array}{cc}1&1\\0&1\end{array}\right)$.

Denoting your two matrices by

$a= \left(\begin{array}{cc}1&1\\1&0\end{array}\right)$,
$b= \left(\begin{array}{cc}2&1\\1&0\end{array}\right)$

we have $a=up$, $b=u^2p$.

Putting $H = \langle a,b \rangle$ and using coset enumeration in Magma, it turns out that $H={\rm GL}(2,\mathbb{Z})$. So your two matrices actually generate all of GL$(2,\mathbb{Z})$.

In fact, denoting $a^{-1}$ and $b^{-1}$ by $A$ and $B$, we have

$p=aBa$, $u=a^2Ba$, $s=abaBAbabA$.

Using the modified coset enumeration algorithm, we can compute a presentation of $H$, which came out as the not particularly enlightening

$H = \langle a,b \mid (aBa)^2, (AbaBA)^2, aBabABAbAbABAbAbABAbAbAB, (abABabaB)^3 \rangle$.