  For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ be the $
k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total
of $k$ terms) and for which each term afterwards is the sum of the $k$
preceding terms. In this paper, for an integer $d\geq 2$ which is square free,
we show that there is at most one value of the positive integer $x$
participating in the Pell equation $x^{2}-dy^{2} =\pm 1$ which is a
$k$--generalized Fibonacci number, with a couple of parametric exceptions which
we completely characterise. This paper extends previous work from [17] for the
case $k=2$ and [16] for the case $k=3$.
