The answer is yes and one can take $$\mathcal{H}:=\mathcal{Hom}_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A,
\mathcal G).$$ Here, $\mathcal F\otimes \mathcal A$ is viewed as a left $\mathcal{End}(\mathcal G)$-module via the isomorphism $\mathcal{End}(\mathcal G)\cong \mathcal{End}(\mathcal F)\otimes \mathcal A$.

More precisely, the right action of $\mathcal{A}$ on $\mathcal F\otimes\mathcal A$
determines a left $\mathcal{A}$-module structure on $\mathcal H=\mathcal{Hom}_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A,
\mathcal G)$, which in trun determines a morphism of $\mathcal O_X$-algebras $\psi:\mathcal A\to \mathcal{End}(\mathcal H)$. This morphism is an isomorphism.

Since both the source and target of $\psi$ are locally free $\mathcal O_X$-modules,
it is enough to check that $\psi$ is an isomorphism after specializing to geometric points. Then the claim follows by noting that the source and target are central simple algebras of the same dimension.

I do feel that something should be said about the use of Morita theory lying in the background, so let me elaborate on this.
I will consider consider the affine case $X=\mathrm{Spec} R$ for simplicity, and write $A=\Gamma(X,\mathcal{A})$, etc.

Our goal is to construct an $(A,R)$-progenerator $H$, i.e., an $(A,R)$-bimodule $H$ such that $H_R$ is projective (i.e. locally free), finitely generated and the natural map $A\to\mathrm{End}_R(H_R)$ is an isomorphism. (In this case, Morita theory tells us that ${}_AH$ is f.g. projective and $R=\mathrm{End}_A({}_AH)$ if $A$-endomorphism are written on the right.)

Write $F'=F\otimes A$ and view it as a right $A$-module.
Then, since $F$ is f.g. projective, there is an $R$-algebra isomorphism
$$
\mathrm{End}_A(F')\cong \mathrm{End}_R(F)\otimes A\cong \mathrm{End}_R(G).
$$
Thus, both $F'$ and $G$ can be regarded as left $\mathrm{End}_R(G)$-modules.

Since $G$ is f.g. projective over $R$ and $F'$ is f.g. projective over $A$,
we see that
$G$ is an $(\mathrm{End}(G),R)$-progenerator and $G$ is an $(\mathrm{End}_A(F'),A)$-progenerator, which we view as a $(\mathrm{End}_R(G),A)$-progenerator. Now, Morita theory tells us that
$H=\mathrm{Hom}_{\mathrm{End}_R(G)}(F',G)$, which is naturally an $(A,R)$-bimodule, is an $(A,R)$-progenerator, which is exactly what we want.