One can find this in section 9.2 of Kempf's book "Algebraic Varieties".

The slightly more general case where $X, Y$ are over an affine scheme $Spec R$ and $\mathcal{F}, \mathcal{G}$ are quasi-coherent sheaves flat over $Spec R$ can be found in

Kempf: "Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves" Rocky Mountain J. Math. Volume 10, Number 3 (1980), 637-646. link

One can find this in section 9.2 of Kempf's book "Algebraic Varieties".

The slightly more general case where $X, Y$ are over an affine scheme $\operatorname{Spec} R$ and $\mathcal{F}, \mathcal{G}$ are quasi-coherent sheaves flat over $\operatorname{Spec} R$ can be found in

Kempf: "Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves" Rocky Mountain J. Math. Volume 10, Number 3 (1980), 637-646. link