Prove or disprove that $G_1/H_1 \cong G_2/H_2$

Often times before trying to prove something, it is helpful to see if the result is true for a few simple examples. In this instance, try letting $G_2$ and $H_1$ both be trivial to see that this result will not hold in general.

No your reasoning is incorrect as the comments have already stated.

As a simple counter example take $G_1 = \mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_2 $, $H_1 = 1 \times 1\times \mathbb{Z}_2$ and then have $G_2 = \mathbb{Z}_2$, with $H_2$ trivial and the map being projection onto the first coordinate. Then clearly $G_1 / H_1 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ and is not isomorphic to $G_1 / H_1$.