The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can attach a Hecke algebra $\mathcal{H}(W,S)$ (this is an exercise in Bourbaki's Groupes et algèbres de Lie IV-VI) with generators $T_w$ ($w \in W$) and relations
$$ \begin{align*} T_w T_{w'} & = T_{ww'} \text{ if } \ell(w)+\ell(w') = \ell(ww') \\ T_s^2 & = a_s + b_s T_s \end{align*} $$
Sometimes, these abstract Hecke algebras turn out to be isomorphic to `convolution type Hecke algebras'
$$ \mathcal{H}(G,B) = \text{End}_G(\text{ind}^G_B \Bbb{1}) $$
for a group $G$ and a subgroup $B$ (I am being deliberately vague here), and one therefore gets a functor
$$ \text{Rep}(G) \longrightarrow \text{R-Mod}(\mathcal{H}(W,S)),\quad V \mapsto V^B $$
relating representations of the abstract Hecke algebra and of the group $G$. This is for instance the case if $(W,S)$ comes from a BN-pair (by another Bourbaki exercise).
Best Answer
$PGL_2(\mathbf{Z})$ is a Coxeter group with presentation $\langle s_1,s_2,s_3 \mid s_i^2=(s_1s_2)^3=(s_1s_3)^2=1 \rangle$, i.e. with labels $\{2,3,\infty\}$. To each Coxeter group $W$ with labels in $\{2,3,4,6,\infty\}$ there exists a Kac--Moody $G$ having a (twin) BN-pair with Weyl Group $W$, as constructed by J. Tits. In particular, $PGL_2(\mathbf{Z})$ is the Weyl group of a Kac-Moody group of rank 3.
While the definition of a Kac-Moody group might appear quite technical, there are two ways of handwaving to convey a flavour of what a Kac-Moody group is like:
(i) A semisimple complex Lie algebra has a nice presentation which can be written down from its Dynkin diagram. Using this presentation for more general Dynkin diagrams, one gets an infinite-dimensional complex Lie algebra $L$ generated by some generators $\langle e_i, f_i, h_i\rangle$. Then the associated adjoint complex Kac-Moody group is by definition the subgroup of Aut$(L)$ generated by ad$(t\cdot e_i)$, ad $(t\cdot f_i)$ for $t \in \mathbf C$.
(ii) One can write down a presentation for $SL_n(\mathbf C)$ with generators the elementary matrices and certain relations which can be read off from the group's associated Dynkin diagram. Then a Kac-Moody group can similiarly be written down in terms of generators and relations for more general Dynkin diagrams.
Caprace-Rémy's "Groups with a twin root datum" gives a very nice introduction to these matters.
On the other hand, by Tits' famous classification of spherical buildings there do not exist thick buildings of types $H_3$ or $H_4$, hence no $BN$-pairs of this type, see e.g. this paper by R. Weiss.