Here is an extremely simple example:
Let $X=U^2\subset \mathbb R^2$ be the open unit disc with center the origin $O$ and let $A=U\setminus \{O\}$.
The long exact sequence for the pair $(X,A)$ yields the segment $\cdots \to H_2(X)=0 \to H_2(X,A) \to H_1 (A)=\mathbb Z \to H_1(X)=0 \to\cdots $ the displayed equalities following from $X$ and $A$ being homotopic respectively to a point and a circle. So we obtain from that segment
First result : $H_2(X,A)=\mathbb Z$
On the other hand the quotient space $X/A$ is the Sierpinski space, a two element space with one closed and one open point. Such a space is known to be contractible, so that we now obtain
Second result: $H_2(X/A)=0$
Conclusion: The pair $(X=U^2,A=U\setminus \{O\})$ is an example for which the relative homology does not coincide with the homology of the quotient space.