In this paper, we offer and discuss three efficient structural solutions for the hardware-oriented implementation of discrete quaternion Fourier transform basic operations with reduced implementation complexities. The first solution: a scheme for calculating sq product, the second solution: a scheme for calculating qt product, and the third solution: a scheme for calculating sqt product, where s is a so-called i-quaternion, t is an j-quaternion, and q is an usual quaternion. The direct multiplication of two usual quaternions requires 16 real multiplications (or two-operand multipliers in the case of fully parallel hardware implementation) and 12 real additions (or binary adders). At the same time, our solutions allow to design the computation units, which consume only 6 multipliers plus 6 two input adders for implementation of sq or qt basic operations and 9 binary multipliers plus 6 two-input adders and 4 four-input adders for implementation of sqt basic operation.