is it possible to write it in one equation
Are you asking about the QR algorithm or the QR decomposition?
The factors $Q$ and $R$ can indeed be written using explicit formulas, since the columns of $Q$ are given by the Gram-Schmidt process, and can thus be expressed by (increasingly complicated) functions of the entries of $A$ that are continuous away from $\det A =0$. As adam points out, once you have $Q$, you also have $R$.
The output of the $QR$ algorithm is the spectrum of $A$. It does have a formula, in a sense, given by the characteristic polynomial; an explicit formula for the eigenvalues of $A$ would also be an explicit formula for the roots of all monic polynomial. To my knowledge no such formula exists, and by the Abel-Ruffini theorem it cannot be algebraic.