What are relations of $S_4$, if generators are $a=(12)$, $b=(1234)$? I think, relations could be $a^2=e$, $b^4=e$, $ababab=e$, $b^2 a b^2 a b = b a b^2 a$.
By two first relations and $b^3 = ababa$ (it is result of our relations),  we can present any element that is generated by $a$ and $b$, as a sequence, where there are not more than one consecutive $a$ and not more than two consecutive $b$. By the last relation we can put all squares of $b$ to the beginning of sequence. We can remove  3 or more consecutive $ab$ by third relations. And also we have $ab^2ab^2ab^2ab^2=e$ as result of relations, so we can remove 4 or more of consectutive $ab^2$ from the beginning of sequence.
But, I'm not sure that we need exactly 4 relations, that there are not  3 relations that would be sufficient.

As Derek Holt mentions in the comments you have $S_4 \cong \langle x,y | x^2 = y^4 = (xy)^3 = 1 \rangle$, so for the generators $a = (1\ 2)$ and $b = (1\ 2\ 3\ 4)$ three relations $a^2 = b^2 = (ab)^3 = 1$ suffice.
Let $G = \langle x,y | x^2 = y^4 = (xy)^3 = 1 \rangle$. In $S_4$ you have a normal subgroup $V \cong C_2 \times C_2$ of order $4$, generated by $b^2 = (1\ 3)(2\ 4)$ and $ab^2a^{-1} = (1\ 4)(2\ 3)$.
So consider in $G$ the subgroup $N = \langle z, w \rangle$ where $z = y^2$ and $w = xy^2x^{-1}$. We would expect that $N$ is a normal subgroup and $N \cong C_2 \times C_2$.
For normality, it is clear that $xzx^{-1} = w$, $xwx^{-1} = z$, and $yzy^{-1} = z$. It remains to check that $ywy^{-1} \in N$. For this, using $(yx)^3 = 1$ and $x^2 = 1$ we find that $yxy = xy^{-1}x$ (as suggested in a comment). Then \begin{align*}
ywy^{-1} &= yxy^2xy^3 \\ &= (yxy)(yxy)y^2 \\ &= (xy^{-1}x)(xy^{-1}x)y^2 \\ &= xy^2xy^2 \\ &= wz. 
\end{align*}
Certainly $N$ has $C_2 \times C_2$ as a quotient. Check that $(zw)^2 = 1$ so in fact $N \cong C_2 \times C_2$: \begin{align*} (zw)^2 &= y^2xy^2xy^2xy^2x \\ &= y(yxy)(yxy)(yxy)yx \\ &= y(xy^{-1}x)(xy^{-1}x)(xy^{-1}x)yx \\ &= y(xy^{-3}x)yx \\ &= yxyxyx = (yx)^3 = 1.\end{align*}
Then $G/N$ is generated by $\overline{x}$, $\overline{y}$ with $(\overline{x})^2 = (\overline{y})^2 = (\overline{x}\overline{y})^3 = 1$, so $G/N$ is a quotient of $S_3$.
Conclude $G \cong S_4$.
