  In this paper, we extend the coupled fractional Fourier transform of a
complex valued functions to that of the quaternion valued functions on
$\mathbb{R}^4$ and call it the quaternion coupled fractional Fourier transform
(QCFrFT). We obtain the sharp Hausdorff-Young inequality for QCFrFT and obtain
the associated R\`enyi uncertainty principle. We also define the short time
quaternion coupled fractional Fourier transform (STQCFrFT) and explore its
important properties followed by the Lieb's and entropy uncertainty principles.
