We want to write the intrinsic form of E.L. equations on the tangent bundle. we define the Poincare-Cartan one-form:
$\Theta = \frac{\partial L}{\partial \dot{q}^{i}}d{q}^{i}$ where L in the lagrangian of the system. It can be shown that the E.L. equations can be written in terms of this one-form as:
$\mathcal{L} _{\Gamma} \Theta - dL =0$, where $\mathcal{L}$ is the lie derivative along $\Gamma$, and $\Gamma$ is the vector field on the fiber bundle TQ, obtained by canonical lifting the tangent vector field of the trajectories on the configuration space Q.
My question is: $\Theta$ is a one-form defined on the configuration space Q, does it make sense consider its lie derivative along a field $\Gamma$ defined on the tangent bundle TQ?
$\Theta=\Theta_L$ is a 1-form defined on $TQ$, not $Q$. If you want, the more abstract definition of $\Theta_L$ relies on the fiber-derivative: \begin{align} \Theta_L= (\mathbf{F}L)^*(\theta), \end{align} where $\mathbf{F}L:TQ\to T^*Q$ is the Fiber-derivative map, and $\theta$ is the tautological 1-form on $T^*Q$ (in terms of adapted coordinates $(q^i,p_i)$ on $T^*Q$, this is given by $\theta=p_i\,dq^i$); and the tauotological 1-form already has a well-known coordinate-free definition.
I think part of your confusion arises from the overload of notation for $q^i$ as a coordinate; people often use it to mean the base coordinate functions on $Q$, or its pullback to $TQ$ or $T^*Q$. Anyway, this a special 1-form in that the second set of coefficients vanish: $\Theta_L=\frac{\partial L}{\partial\dot{q}^i}\,dq^i+\sum_{i=1}^n0\,d\dot{q}^i$.
Now, because $\Theta_L$ is a $1$-form on $TQ$, its makes sense to consider its Lie-derivative with respect to a vector field on $TQ$.