The special affine Fourier transform (SAFT) or offset linear canonical transform was introduced by S. Abe and J.T. Sheridan in {\em S. Abe and J.T. Sheridan, Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach, J. Phys. {\bf 27} (12), 4179--4187 (1994)}, which generalizes the Fourier transform, fractional Fourier transform, Fresnel transform, Laplace transform, offset Fourier transform, offset fractional Fourier transform, and linear canonical transform. As it has plenty of applications in signals and systems, there are many research works have been carried out on this transform. In particular, extending the transform to larger class of functions is also one of the interesting research problem on SAFT. In this direction, recently the special affine Fourier transform is extended to the context of two dimensional quaternion valued functions, and proved its properties including various uncertainty principles in {\em Y. El Haoui, S. Fahlaoui, E. Hitzer, Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform, arXiv:1807-04068v3 [math.CA] 18 September 2019}. In this paper, using the idea employed in {\em R. Roopkumar, Quaternionic one-dimensional fractional Fourier transform, Optik, Vol. 127 (2016), 11657-11661}, we define a one-dimensional quaternionic special affine Fourier transform and prove the following properties: \begin{enumerate} \item Linearity over quaternions. \item Inversion formula. \item Parseval's identiy. \item Dilation property. \item Shifting properties. \item Modulation property. \item Uncertainty properties. \end{enumerate} Following {\em A. Bhandari and A. I. Zayed, “Convolution and product theorems for the special affine Fourier transform,” in Frontiers in Orthogonal Polynomials and q-Series. M. Z. Nashed and X. Li, Eds. Singapore: World Scientific, 119--137 (2018)}, we also introduce a convolution of quaternion valued functions, using which we get the convolution and product theorems for the extended one-dimensional QSAFT.