I would say no; the example is when $R$ is a hyperbolic plane, the relation you demand says merely that $Q_1$ and $Q_2$ are in the same genus. This is in SPLAG, page 378 in the first (1988) edition, see also Clark Jagy.
Give me a few minutes, I am going to display integer equivalence for $Q_1 = x^2 + 14 y^2,$ $Q_2 = 2 x^2 + 7 y^2,$ which pair take up a good deal of room in Cox's book, and $R = 2 z w.$ By solving $P^T A_1P = A_2$ in 4 by 4 integer matrices...