Let \(G=\mathrm{SO}(3,1)\) and \(T\) be the left quasi-regular representation of \(G\) acting on the space \(\mathfrak{D}\) of \(\sigma\)-homogeneous infinitely differentiable functions on the semicone \(x_1^2-x_2^2-x_3^2-x_4^2=0, x_1>0\). \(\mathfrak{D}\) has two types of bases. One is the spherical basis expressed by the Gegenbauer polynomials and the other is the parabolic basis expressed by the Bessel functions of the first kind. In their previous paper [J. Anal. 31, No. 1, 719–732 (2023; Zbl 1524.33032)] the authors proved that the matrix coefficients of \(T(a)\), \(a=\mathrm{diag}(1,1,1,-1)\), with respect to the parabolic basis are written in terms of hyper Bessel-Clifford functions. In this paper they calculate the matrix coefficients of \(T(e)\) with respect to spherical and mixed bases. Then, considering the continual additional theorem of the matrix coefficients, they obtain a new formula for evaluating an integral containing hyper Bessel-Clifford functions in terms of Meijer’s \(G\)-functions. Similarly, from the countable addition theorem, a new relation between hyper Bessel-Clifford functions and Meijer’s \(G\)-functions follows.