The assertion if false.

Suppose, for example, that both G and N₁ are the additive group Z of integers. Let N₂ be the additive group 2Z of even integers.

Then G/N₁ = Z/Z is the trivial group and G/N₂ = Z/2Z is the group of two elements. Clearly, G/N₁ and G/N₂ are not isomorphic in this case.