Gale and Shapley barely make any assumptions about preferences. They don't need a functional form, simply an ordinal ranking of the options. Moreover, there are no transfers in this setting, only the discrete matching options.
Even Kelso and Crawford do not use quasi-linear utility. They assume some utility function that is increasing and continuous in salary, but not necessarily linearly increasing.
In Shapley and Shubik (1971) monetary transfers enter linearly.
However, I wouldn't call this quasi-linear utility, because in quasi-linear utility most people have the following functional form over a good $x$ and money $m$ in mind: $u(x,m) = v(x) - m$, where $v$ is a continuous function.
In their model, the $v(x)$ is simply the money value of house $x$, where the houses are district options.