Let me give some context for the question. Following Lurie's convention, by an $\infty$-category/quasicategory I mean a simplicial set $S:\Delta^{op}\to Set$ satisfying the inner horn filling condition, that is, for all $0<k<n$, any map $\Lambda^n_k\to S$ can be extended to a map $\Delta^n\to S$ along the inclusion $\Lambda^n_k\hookrightarrow \Delta^n$.
We can define a relation on the edges of $S$, i.e. $\phi:\Delta^1\to S$ (equivalently $\phi \in S_1$) as follows. For an edge $\phi : \Delta^1 \to S$ we use the notation $\phi : C\to C'$ to mean that $d_1\phi = C$ and $d_0\phi = C'$. Then, given two edges $\phi, \phi' : C\to C'$ we say they are homotopic if there is a 2-simplex $\sigma : \Delta^2 \to S$ with $d_0\sigma = 1_{C'}:=s_0C'$, $d_1\sigma = \phi'$ and $d_2\sigma = \phi$. Let's denote this relation $\sim_1$. (There's a slight chance I messed up the indexes here, hence I attached a picture to state the correct diagram generally used to represent the 2-simplex). 
It turns out this is an equivalence relation. But we could very much have defined the relation by "swapping" $\phi$ and $1_{C'}$, i.e. we could have said : $\phi\sim_2 \phi'$ if and only if there is a 2-simplex $\sigma : \Delta^2 \to S$ with $d_0\sigma = \phi$, $d_1\sigma = \phi'$ and $d_2\sigma = 1_C$. Since a picture is worth a thousand words, we can describe this with the following diagram :
(In the picture the name of the edges and vertices are changed because I simply took a screenshot of a paper by Moritz Groth which didn't use the same naming convention as I did in the first picture)
As it turns out, when $S$ is a quasicategory, these two relations coincide, that is, $\phi \sim_1\phi' \iff \phi\sim_2 \phi'$. I tried proving these two were indeed equivalent. My goal was to start with say a homotopy in the first sense, this gives a 2-simplex. Cook up two other to specify a map $\Lambda_k^3\to S$ for some $0<k<3$, which would allow me to exhibit a homotopy in the second sense as the new face of the 3-simplex obtained with the horn filling condition. I've tried various combinations of 2-simplices but I couldn't find the one that gives me what I needed.
You can find a proof in Rezk's notes here as proposition 11.6.
Briefly, here is the strategy. Write $f \sim_l f'$ if there exists a $2$-simplex showing $f \simeq f' \circ 1$, and $f \sim_r f'$ if there exists a $2$-simplex showing $1 \circ f \simeq f'$. Then we shall show:
The two statements combined imply that the equivalence relations $\sim_l$ and $\sim_r$ are symmetric, and also equal.
You can find the $3$-simplices that establish these statements on the bottom of page 30 in Rezk's notes. To make this post self-contained, here are images of these diagrams. In this first diagram, we have a $\Lambda^3_1$ horn where the left triangle witnesses $f \sim_l f'$ and the other two right triangles are degeneracies. Filling in the simplex, we obtain the outer triangle, which is a witness for $f \sim_r f'$.
In the second diagram, we have a $\Lambda^3_2$ horn where the upper right triangle witnesses $f \sim_r f'$ and the other two triangles are degeneracies. Filling in the simplex, we obtain the outer triangle, which is a witness for $f' \sim_l f$.