The differential of Φ$\Phi$ at a point (D,θ)$(D,\theta)$ in the fiber product is given by the induced map on tangent spaces:
T(Div≥0(X))×T(Pic(X))×T(Aut0(X)) → T(G(n,|L|))
$$T(\mathrm{Div} _{\ge 0}(X)) \times T(\mathrm{Pic} (X)) \times T(\mathrm{Aut} ^0 (X)) → T(\mathbb G (n,|\mathcal L|))$$
where T$T$ denotes the tangent space.
To compute this differential, we can use the fact that the tangent space of Div≥0(X)$\mathrm{Div} _{\ge 0}(X)$ at a point D$D$ is isomorphic to H0(X,OD)$H^0(X,\mathcal O _D)$, the space of global sections of the structure sheaf of X$X$ twisted by D$D$. Similarly, the tangent space of Pic(X)$\mathrm {Pic} (X)$ at a point L$\mathcal L$ is isomorphic to H1(X,OL)$H^1 (X, \mathcal O _{\mathcal L})$, the space of global sections of the sheaf of holomorphic sections of L$\mathcal L$. Finally, the tangent space of Aut0(X)$\mathrm{Aut} ^0 (X)$ at the identity is isomorphic to H0(X,TX)$H^0 (X,TX)$, the space of vector fields on X$X$.
Using these identifications, we can write down the differential of α, β$\alpha$ and $\beta$, and the map induced by Q⊆|M|$Q \subseteq |\mathcal M|$ on the space of global sections of L$\mathcal L$. The differential of α$\alpha$ is given by the map
T(Div≥0(X)) → T(Pic(X)), f ↦ OX(f+D) ⊗ OX(D)^{-1}
$$T(\mathrm{Div} _{\ge 0}(X)) \ni f \mapsto \mathcal O _X (f+D) \otimes \mathcal O _X (D)^{-1} \in T(\mathrm{Pic} (X))$$
while the differential of β$\beta$ is given by the map
T(Aut0(X)) → T(Pic(X)), v ↦ L ⊗ v^* M^{-1} ⊗ M ⊗ L^{-1}.
$$T(\mathrm {Aut} ^0 (X)) \ni V \mapsto \mathcal L \otimes V^* \mathcal M ^{-1} \otimes \mathcal M \otimes {\mathcal L} ^{-1} \in T(\mathrm{Pic} (X)) \ . $$
To compute the differential of the map induced by Q$Q$, we need to consider the linear system Q$Q$ as a subspace of H0(X,M)$H^0 (X, \mathcal M)$. Let s_1,...,s_n$s_1, \dots, s_n$ be a basis for Q$Q$, and let t_1,...,t_m$t_1, \dots, t_m$ be a basis for H0(X,M)$H^0 (X, \mathcal M)$ such that s_1,...,s_n,t_1,...,t_m$s_1, \dots, s_n, t_1, \dots, t_m$ is a basis for H0(X,M)$H^0 (X, \mathcal M)$. Then the map induced by Q$Q$ is given by the matrix
[ s_1,...,s_n,t_1,...,t_m ].
$$[ s_1, \dots, s_n, t_1, \dots, t_m ] \ .$$
To compute the differential of Φ$\Phi$, we use the product rule for differentials:
dΦ(D,θ) = d(D+θ^*Q) + (dα(D),dβ(θ)) + (d(Q),0),
$$\mathrm d \Phi (D, \theta) = \mathrm d (D + \theta^* Q) + (\mathrm d \alpha(D), \mathrm d \beta(\theta)) + (\mathrm d (Q),0) \ ,$$
where d(Q)$\mathrm d (Q)$ is the differential of the map induced by Q$Q$. The first term on the right-hand side is just the differential of the translation map T_Div≥0(X)$T_{\mathrm {Div} _{\ge 0} (X)}$ given by d(D+θ^Q) = dD + θ^ dQ$\mathrm d (D + \theta \wedge Q) = \mathrm d D + \theta \wedge \mathrm d Q$, while the second term is the product of the differentials of α$\alpha$ and β$\beta$. The last term is zero, since the differential of Q$Q$ does not depend on θ$\theta$.
Putting everything together, we obtain the differential of Φ$\Phi$ at a point (D,θ)$(D, \theta)$ in the fiber product.
