The answer is no.  The Deletion Condition says that any expression in the generators of a Coxeter group contains a reduced expression as a subexpression.  Since $a$ and $b$ don't commute, the unique reduced expression for $ab$ is $ab$, which must therefore occur as a subexpression of any longer expression that evaluates to $ab$, i.e. any such expression must contain an $a$ before a $b$.