Mathematical analysis of an economic model - stuck in calculations
There exist implicit assumptions in the set up that exclude various cross-price effects on the demand side, related to business behavior as well as to consumer behavior. Accepting the set up (and assuming downward-sloping demand and upward-sloping supply), the old-fashioned verbal treatment goes as follows:  

An increase in $G$ will move the demand schedule $Q_1^d$ outwards, in the $\{Q_1, P_1\}$ space. This will tend to bring a new equilibrium for good 1, at a higher price $P_1$, and a higher quantity $Q_1$.

A higher $P_1$ increases costs for the supply of good $2$, and so it will shift upwards the supply curve for good 2 in the $\{Q_2, P_2\}$ space, while leaving the demand schedule for good $2$ unaffected. This will produce an equilibrium with higher $P_2$ and lower $Q_2$. A higher $P_2$ in turn will tend to shift upwards the supply curve for good 1 in the $\{Q_1, P_1\}$ space (again, because due to higher costs, lower quantity is supplied at any own price than previously).

This will tend to increase equilibrium $P_1$ even more, but it will tend to offset the increase in the new equilibrium $Q_1$.  

So we have learned:
In market $2$, an increase in $G$ for good $1$ will lead to an equilibrium with higher $P_2$ and lower $Q_2$.
In market $1$ an increase in $G$ will lead to an equilibrium with higher $P_1$, but ambiguous effect on the new equilibrium $Q_1$.

I'll leave to the OP the task of mathematizing these results and see also whether the ambiguity can be resolved.