Calculus and Indifference Curves in an Urban Economics Example
The utility function under consideration is $v(c,q)$ and then
$$MRS(c,q) = \frac{\partial v/\partial q}{\partial v/\partial c} = v_2/v_1$$
make the functional denpendency of on $u$ explicit then you have
$$\frac{\partial}{\partial u}MRS(c(u),q(u)) = \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
where the last identity follows because you know that $p = MRS$. Now simply rearrange the identity
$$\frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
to get
$$ \frac{\partial q(u)}{\partial u} = \frac{\left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right]}{\frac{\partial MRS(c(u),q(u))}{\partial q} },$$
then use that Brueckner has defined $\eta := \left[\frac{\partial MRS(c(u),q(u))}{\partial q}\right]^{-1}$ in footnote(3) to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right] \eta ,$$
and finally apply the rule that $\frac{\partial c(u)}{\partial u} = \frac{1}{\partial v/\partial c} = 1/v_1$ to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{1}{v_1}\right] \eta.$$