\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
At each \(q\in L\cap Q\), consider the Lagrangian submanifold \(T^*qQ\).
Take local coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\) identifying \(q\) with the origin so that \(T^*qQ\) is the linear subspace in the \(p_i\) directions.
We can take a Weinstein neighborhood \(T^*_\epsilon(T^*_qQ)\) of \(T^*_qQ\), whose cotangent bundle structure is
\begin{align*}
T^*_\epsilon(T^*_qQ)\to T^*qQ && (q_1, \ldots, q_n, p_1, \ldots, p_n)\mapsto (p_1, \ldots, p_n).
\end{align*}
Since the intersection \(L\cap Q\) is transverse, the projection \(T_qL\to T_q(T^*qQ)\) is surjective. Therefore when restricted to a small enough neighborhood of \(\in U\subset T^*_qQ\) the Lagrangian \(L|_{T^*_\epsilon U}\) presents itself as a section of \(T^*_\epsilon U\to U\).
Therefore, there exists a one form \(\eta\in \Omega^1(U)\) so that \(L|_{T^*U}\) is parameterized by \((p, \eta_p)\).
By taking an even smaller \(U\), we may assume that \(U\) is a contractible neighborhood, and \(\eta=dH\) is an exact one-form.
Pick \(\rho\) a function which vanishes in a neighborhood of \(q\in U\), and takes the value \(1\) in a neighborhood of \(\partial U\).
Consider the Lagrangian section of \(T^*U\) parameterized by \(d(\rho \cdot H)\).
This section is Hamiltonian isotopic to \(L|_{T^*_\epsilon U}\) relative boundary.
Additionally, \(d(\rho\cdot H)\) agrees with \(U=T^*_qQ\) in a small neighborhood of \(q\).
The Lagrangian submanifold \(L\setminus (L|_{T^*U})\cup (d(\rho\cdot H))\) is Hamiltonian isotopic to \(L\).