I did more computation recently, and I got what I desired.

Firstly, to be exact, it should be the cohomology group rather than homology group, and presentation in the question is wrong, it should be $$\Bbbk\left<X_i,\partial_j\right>_{{1\leq i\leq n}\atop{1\leq j\leq n-1}}\bigg/ \left<\begin{array}{c} \partial_i\partial_{i-1}\partial_i=\partial_{i-1}\partial_i\partial_{i-1},\\ |i-j|\geq 2, \quad \partial_i\partial_j=\partial_j\partial_i,\\ \partial_i^2=0. \end{array}\begin{array}{c} X_iX_j=X_jX_i,\\ \partial_iX_j-X_{s_i(j)}\partial_i\\ =\delta_{i,j}-\delta_{i+1,j}. \end{array}\right>$$ I was mislead by Kumar's definition (Kac-Moody Groups, their Flag Variety and Representation Theory) of Nil-Hecke ring and the definition of convolution in homology.

• To prove this, one can first do it in nonequivairant case, the $G$-orbits of $G/B\times G/B$ are one-to-one correspondent to $B$-orbit of $G/B$, i.e. Schubert cells.
• The Poincar'e duality of each, say $\partial_w$, with respect to the Schuber cell $BwB/B$, acts on $H^*(G/B)$ by the Demazure operator $\partial_w$. To check this, it suffices to do the intersection product, where they all intersects transversally.
• The $X_i=X_i\partial_e$, where $H^*(G/B)$ acts on $H^*(G/B\times G/B)$ by the first projection acts on $H^*(G/B)$ by left multiplication $X_i$.
• Now relation is easy to check, by a standard topological argument, it is an isomorphism (for example, Harish–Leray). Actually, $H^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{\Bbbk}(H^*(G/B))$.
• To deal with equivariant case, we first do it in $T$-equivariant case, it is harmless since $H_G^*(X)\to H_T^*(X)$ is always injective ($\operatorname{char} \Bbbk=0$).
• There no longer exists Poincar'e duality, but the pairing of cells also gives a well-defined cohomology class. The computation of the result of paring of cells in nonequivariant case can be directly move to equivariant case. As a result, so it also acts as Demazure operator.
• The rest is completely the same to nonequivariant case. Actually, $H_G^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{H_G^*(pt)}(H^*(G/B))$. The actions are all $H_G^*(pt)$ map by the associativity of convolution $$H_G^*(G/B\times G/B)\stackrel{\displaystyle\curvearrowright}{\phantom{\square}} H_G^*(G/B\times pt)\stackrel{\displaystyle\curvearrowleft}{\phantom{\square}} H_G^*(pt\times pt). $$

I did more computation recently, and I got what I desired.

Firstly, to be exact, it should be the cohomology group rather than homology group, and presentation in the question is wrong, it should be $$\Bbbk\left<X_i,\partial_j\right>_{{1\leq i\leq n}\atop{1\leq j\leq n-1}}\bigg/ \left<\begin{array}{c} \partial_i\partial_{i-1}\partial_i=\partial_{i-1}\partial_i\partial_{i-1},\\ |i-j|\geq 2, \quad \partial_i\partial_j=\partial_j\partial_i,\\ \partial_i^2=0. \end{array}\begin{array}{c} X_iX_j=X_jX_i,\\ \partial_iX_j-X_{s_i(j)}\partial_i\\ =\delta_{i,j}-\delta_{i+1,j}. \end{array}\right>$$ I was mislead by Kumar's definition (Kac-Moody Groups, their Flag Variety and Representation Theory) of Nil-Hecke ring and the definition of convolution in homology.

• To prove this, one can first do it in nonequivairant case, the $G$-orbits of $G/B\times G/B$ are one-to-one correspondent to $B$-orbit of $G/B$, i.e. Schubert cells.
• The Poincar'e duality of each, say $\partial_w$, with respect to the Schuber cell $BwB/B$, acts on $H^*(G/B)$ by the Demazure operator $\partial_w$. To check this, it suffices to do the intersection product, where they all intersects transversally.
• The $X_i=X_i\partial_e$, where $H^*(G/B)$ acts on $H^*(G/B\times G/B)$ by the first projection acts on $H^*(G/B)$ by left multiplication $X_i$.
• Now relation is easy to check, by a standard topological argument, it is an isomorphism (for example, Harish–Leray). Actually, $H^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{\Bbbk}(H^*(G/B))$.
• To deal with equivariant case, we first do it in $T$-equivariant case, it is harmless since $H_G^*(X)\to H_T^*(X)$ is always injective ($\operatorname{char} \Bbbk=0$).
• There no longer exists Poincar'e duality, but the pairing of cells also gives a well-defined cohomology class. The computation of the result of paring of cells in nonequivariant case can be directly move to equivariant case. As a result, so it also acts as Demazure operator.
• The rest is completely the same to nonequivariant case. Actually, $H_G^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{H_G^*(pt)}(H^*(G/B))$. The actions are all $H_G^*(pt)$ map by the associativity of convolution $$H_G^*(G/B\times G/B)\stackrel{\displaystyle\curvearrowright}{\phantom{\square}} H_G^*(G/B\times pt)\stackrel{\displaystyle\curvearrowleft}{\phantom{\square}} H_G^*(pt\times pt). $$

The last two points are wrong. The real reason is, over characteristic zero case, $H_G(G/B)$ is known to be free of rank $\dim H(G/B)$ over $H_G(pt)$. So bthe convolution algebra is eaxctly of rank $\dim H(G/B)^2$ over $H_G(pt)$. This is the main point.