First, I am very skeptical that one can, in general, calculate $H_2(G, \mathbb{Z})$ using  **just**
a presentation $\langle a,b \, \vert \, r_1=r_2 =1 \rangle$.  It is generally known that finite presentation for 
groups are inadequate to solve many kinds of decision problems.  You will probably need to know
**more** about $G$ to calculate  $H_2(G, \mathbb{Z})$ using either the Hopf formula or 
 spectral sequences.  

Second, if you do know of one example of a group $G$ which has a presentation 
$\langle a,b \, \vert \, r_1=r_2 =1 \rangle$  and such that 
$H_2(G, \mathbb{Z})$, is $\mathbb{Z}\times \mathbb{Z}$, then it is not difficult to construct lots of other 
presentations for $G$.  Let $F$ be the free group on $\{a,b\}$.  You can use a sequence of regular  
elementary Nielsen transformations (See Lyndon and Schupp, Chapter I,  Section 2) to find another 
pair of words, $\{A,B\}$ that also freely generate $F$.  Write $R_i, i=1,2$ for the word obtained from $r_i$ 
by replacing every $a$ by $A$ and every $b$ by $B$ (and doing the same replacement for inverses, 
of course).  Then   $\langle A,B \, \vert \, R_1=R_2 =1 \rangle$ is also a presentation for $G$.  Then 
$a$ can be written as a word on $A$ and $B$ and $b$ can be written as a word on $A$ and $B$, so we can 
write $r_1$ and $r_2$ as words on $A$ and $B$ and obtain a presentation 
$\langle A,B \, \vert \, r_1=r_2 =1 \rangle$ for $G$.  Dually, we can rewrite $R_1$ and $R_2$ as words on 
$a$ and $b$ and obtain a presentation $\langle a,b \, \vert \, R_1=R_2 =1 \rangle$.  Similarly, 
we could obtain a new presentation for $G$ by swapping $a$ with $b$ in $r_1$ and $r_2$.  Also we 
can replace $r_1$ and 
$r_2$ by conjugates of these.   

Finally, I will mention one example which many might consider to be obvious.  Let $G$ be the 
 group on 2 generators which is free in the variety of groups having nilpotence class 2.  Then $G$ 
 has numerous presentations of the form $\langle a,b \, \vert \, r_1=r_2 =1 \rangle$ and the 
 Schur multiplier, $H_2(G, \mathbb{Z})$, is $\mathbb{Z}\times \mathbb{Z}$.  
 
 Write $\{\gamma_k(H)\}$ for the lower central series of any group $H$:  i,e. 
 $\gamma_1(H) = H$ and $\gamma_{k+1}(H) = [\gamma_k(H),H]$.  Let $F$ be the free group on two 
 generators, $a$ and $b$. Let $G$, as above,  be the 
 group on 2 generators which is free in the variety of groups having nilpotence class 2.  Then 
 $G = F/\gamma_3(F)$ and one presentation for $G$ is $\langle a,b \, \vert \, [b,a,a]=[b,a,b]=1\rangle$.
 Here, the smallest normal subgroup, $R$, containing $[b,a,a]$ and $[b,a,b]$ is $\gamma_3(F)$. 
  It is not too difficult to see that $F/\gamma_4(F)$ is the covering group 
 for $G$ and that $\gamma_3(F)/\gamma_4(F)$ is the Schur multiplier.  It is also easy to verify that  $\gamma_3(F)/\gamma_4(F)$ is free abelian with generators $[b,a,a]\gamma_4(F)$ and 
$[b,a,b]\gamma_4(F)$.