Let ${X\_1 ,\ldots ,X\_m }$ be a basis of the space of horizontal vector fields on the Carnot group $\mathbb{G}=(\mathbb{R}^N, \circ)(m < N)$. We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem \begin{align\*} &\int\limits\_\Omega\\! {\langle \langle AXu,Xu\rangle ^{\frac{p-2}{2}}AXu,X(v-u)\rangle } dx\\\ &\quad\geq\int\limits\_\Omega\\! {B(x,u,Xu)(v-u)}dx +\int\limits\_\Omega \\!{\langle f(x),X(v-u)\rangle } dx,\quad \text{for all } v\in \mathcal{K}\_\psi^\theta (\Omega), \end{align\*} where $A=(a\_{ij}(x)){m\times m}$ is a symmetric positive-definite matrix with measurable coefficients, $p$ is close to 2, $\mathcal{K}\psi ^\theta(\Omega )={v\in HW^{1,p}(\Omega)\colon v\geq \psi ;\mbox{a.e.};{\rm in};\Omega ,v-\theta\in HW\_0^{1,p} (\Omega )}$, $\psi$ is a given obstacle function, $\theta$ is a boundary value function with $\theta \geq \psi$ . We first prove the $C\_X^{0,\alpha}$ regularity of solutions provided that the coefficients of $A$ are of vanishing mean oscillation (VMO). Then the $C\_X^{1,\alpha }$ regularity of solutions is obtained if the coefficients belong to the class $\mbox{BMO}\_\omega$ which is a proper subset of VMO.