  A Hecke endomorphism algebra is a natural generalisation of the $q$-Schur
algebra associated with the symmetric group to a Coxeter group. For Weyl
groups, B. Parshall, L. Scott and the first author \cite{DPS,DPS4} investigated
the stratification structure of these algebras in order to seek applications to
representations of finite groups of Lie type. In this paper we investigate the
presentation problem for Hecke endomorphism algebras associated with arbitrary
Coxeter groups. Our approach is to present such algebras by quivers with
relations. If $R$ is the localisation of $\mathbb Z[q]$ at the polynomials with
the constant term 1, the algebra can simply be defined by the so-called
idempotent, sandwich and extended braid relations. As applications of this
result, we first obtain a presentation of the 0-Hecke endomorphism algebra over
$\mathbb{Z}$ and then develop an algorithm for presenting the Hecke
endomorphism algebras over $\mathbb Z[q]$ by finding torsion relations. As
examples, we determine the torsion relations required for all rank 2 groups and
the symmetric group $\mathfrak{S}_4$.
