It is written in the book The Physics of Graphene (Page 10 and 17) that when the intervalley scattering is neglected, we can make the following substitution in the Hamiltonian of the graphene when using the tight-binding model: $$\hbar q_x\to\hat{p}_x=-i\hbar\frac{\partial}{\partial x}$$ $$\hbar q_y\to\hat{p}_y=-i\hbar\frac{\partial}{\partial y} $$ Then the effective Hamiltonian goes from $$\hat{H}_{\mathit{K},\mathit{K'}}(\vec{q})=\hbar v \begin{pmatrix} 0 & q_x \mp iq_y\\ q_x \pm iq_y & 0 \end{pmatrix}$$ to $$\hat{H}_{\mathit{K}}=-i\hbar v\vec{\sigma}\nabla$$ $$\hat{H}_{\mathit{K'}}=\hat{H}_{\mathit{K}}^{\text{T}}$$ where $\vec{q}=\vec{k}-\vec{\mathit{K}}$ denotes the relative momentum in the vicinity of the Dirac point $\mathit{K}$; $v=\frac{3a|t|}{2\hbar}$ is the electron velocity at the conical points; $\vec{\sigma} = (\sigma_x, \sigma_y)$ are the pauli matrices and T denotes a transposed matrix.
It hints that it corresponds to the effective mass approximation, or $\vec{k}\cdot\vec{p}$ perturbation theory, however, I still do not get the idea why this can be done.