Different aspects of Berezin quantization are investigated for the Heisenberg motion group, defined as the semi-direct product of the Heisenberg group \(H_n\) with a compact subgroup \(K\) of the unitary group \(U(n)\). Berezin quantization of homogeneous manifolds \(M=G/H\) via coherent states has been enlarged with the Stratonovich-Weyl correspondence for the triple \((G,\pi,M)\), where \(\pi\) is a unitary representation of the Lie group \(G\) on a Hilbert space \(\mathcal{H}\) [H. Figueroa et al., J. Math. Phys. 31, 2664–2671 (1990; Zbl 0753.43002)]. The Heisenberg motion group is a group of Harish-Chandra type, and the author applies the method developed in Chapter XII in the book of K.-H. Neeb [Holomorphy and convexity in Lie theory. Berlin: de Gruyter (1999; Zbl 0936.22001)] for obtaining unitary irreducible representations on some generalized Fock space. The simplest example is the case of the Heisenberg group, when the usual Weyl correspondence provides a Stratonovich-Weyl correspondence for the triple \((H_n,\sigma_0,\mathbb{R}^{2n})\) for the Schrödinger model on \(L^2(\mathbb{R}^n)\). The Bargmann-Fock representation \(\pi_0\) is defined on the Fock space \(\mathcal{F}_0=L^2(\mathbb{C}^n,\mu_{\lambda})\) of square integrable holomorphic functions of norm \(||F||^2_{\mathcal{F}_0}=\int_{\mathbb{C}^n}|F(z)|^2\text{e}^{-\frac{|z|^2}{2\lambda}} d\mu_{\lambda}<\infty\), where \(\lambda>0\) indexes the Schrödinger representation of \(H_n\) or the orbits \(\mathcal{O}_{\lambda}\subset\mathfrak{h}_{n}^*\). Here \(d\mu_{\lambda}=(2\pi\lambda)^{-n}d \operatorname{Re}(z) d\operatorname{Im}(z)\), \(z\in\mathbb{C}^n\), and \(\mathfrak{h}_n\) denotes the Lie algebra of the Heisenberg group \(H_n\). The Segal-Bargmann transform \(B_0:L^2(\mathbb{R}^n)\rightarrow \mathcal{F}_0\) is applied. The Berezin transform is the operator \(\mathcal{B}_0=S^0(S^0)^*\) defined on \(\mathcal{F}_0\), where \(S^0\) defines the Berezin symbol. The Berezin symbol \(S^0\) is transfered to operators on \(L^2(\mathbb{R}^n)\). Two Stratonovich-Weyl maps are constructed for \((H_n,\sigma_0,\mathcal{O}_{\lambda})\) and \((H_n,\pi_0,\mathcal{O}_{\lambda})\). The case of the Jacobi group \(G^J_n=H_n\rtimes \mathrm{Sp}(n,\mathbb{R})\), also a group of Harish-Chandra type, was investigated in [the author, Rend. Semin. Mat. Univ. Padova 136, 6–93 (2016; Zbl 1359.22010)]. A preliminary investigation of the Stratonovich-Weyl correspondence via Berezin quantization applied to the Heisenberg motion group was done in [the author, Rend. Ist. Mat. Univ. Trieste 46, 157–180 (2014; Zbl 1322.22014)]. Generalities about the Heisenberg motion group are extracted from the paper [the author, Riv. Mat. Univ. Parma (N.S.) 4, No. 1, 197–213 (2013; Zbl 1293.22003)], devoted to the \((2n+2)\)-dimensional real diamond (oscillator) group. Applying the decomposition of the complexification of the Lie algebra of the Heisenberg motion group as group of Harish-Chandra type, the domain corresponding to the \(\mathfrak{p}^+\) is denoted by \(\mathcal{D}\approx\mathbb{C}^n\). A unitary irreducible representation \(\rho\) of \(K\) on a finite dimensional Hilbert space is fixed. The Hilbert space \(\tilde{\mathcal{F}}\) of holomorphic functions on \(\mathcal{D}\) with values in \(V\) is introduced, after the \(G\)-invariant measure on \(\mathcal{D}\) is fixed, where now \(G\) denotes the Heisenberg motion group. A holomorphic representation \(\tilde{\pi}\) of \(G\) on the Hilbert space \(\tilde{\mathcal{F}}=\mathcal{F}_0\otimes V\) is constructed, which is replaced with an equivalent representation \(\pi \), whose restriction to \(H_n\) is precisely \(\pi_0\). A Fock model for the Heisenberg motion group is introduced, together with the associated Berezin calculus. Next, the corresponding generalized Segal-Bargmann transform \(B:L^2(\mathbb{R}^n,V)\rightarrow \mathcal{F}\) is introduced. The last chapter is devoted to the establishing of the Stratonovich-Weyl correspondence via the Weyl calculus for the Fock model of the Heisenberg motion group.