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 presentations 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 $[b,a,b]\gamma_4(F)$.

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 presentations 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 $[b,a,b]\gamma_4(F)$.

Later (November 16, 2015): For another example for $G$, let $G$ be the group having presentation $\langle a,b \, \vert \, [b,a,a] = [b,a,b,b]=1 \rangle$.

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 $[b,a,b]\gamma_4(F)$.

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 presentations 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 $[b,a,b]\gamma_4(F)$.