  An algorithm is constructed that, when given an explicit presentation of a
finitely generated nilpotent group $G,$ decides for any pair of endomorphisms
$\varphi, \psi : G \to G$ and any pair of elements $u, v \in G,$ whether or not
the equation $(x\varphi)u = v (x\psi)$ has a solution $x \in G.$ Thus it is
shown that the problem of the title is decidable. Also we present an algorithm
that produces a finite set of generators of the subgroup (equalizer)
$Eq_{\varphi, \psi}(G) \leq G$ of all elements $u \in G$ such that $u \varphi =
u \psi .$
