The authors construct generalized \(\vartheta\)-functions on the Kodaira-Thurston manifold \(M\) [K. Kodaira, ”On the structure of compact complex analytic surfaces. I.” Am. J. Math. 86, 751–798 (1964; Zbl 0137.17501)]. \(M\) is a compact, 4-dimensional nilmanifold which is symplectic and complex, but not Kähler [W. P. Thurston, Proc. Am. Math. Soc. 55, 467–468 (1976; Zbl 0324.53031)]. This construction is relevant in the context of the almost Kähler quantization [D. Borthwick and A. Uribe, Math. Res. Lett. 3, No. 6, 845–861 (1996; Zbl 0872.58030)]. Here \(M:=\Gamma\backslash G\), where \(G=\text{Heis}(3)\times\mathbb{R}\), \(\text{Heis}(3)\) is the real 3-dimensional Heisenberg group, while \(\Gamma\) is the integer lattice in \(G\). The left-invariant symplectic form on \(G\) descends to a symplectic \(\omega\) form on \(M\). The Hamiltonians of right-invariant vector fields form a Lie algebra \(\tilde{\mathfrak{g}}\), the central extension of the Lie algebra \(\mathfrak{g}\) of \(G\). The corresponding simply connected Lie group is \(\tilde{G}=G\rtimes\mathbb{R}\) such that \(\tilde{G}\) acts on \(M\) in a Hamiltonian fashion. The principal circle bundle \(P:=\tilde{\Gamma}\backslash\tilde{G}\rightarrow M\), where the lattice \(\tilde{\Gamma}\) covers \(\Gamma\), has a natural connection whose curvature is the symplectic form \(\omega\) on \(M\) and let \( \ell\rightarrow M\) denote the associated prequantization line bundle. There is a right quasiregular representation \(\rho\) of \(\tilde{G}\) on \(L^2(M,\ell)\). It is shown that \(\rho\) decomposes into a direct sum of unitary irreducible representations \(\pi_k : \tilde{G}\rightarrow\text{End}(V_k)\), as \(L^2(M,\ell^{\otimes k})\simeq 4k^2V_k\). Each representation spaces \(V_k\) is isomorphic to \(L^2(H\backslash\tilde{G})\), where \(H\) is any choice of a certain family of subgroups of \(\tilde{G}\), those with Lie algebra is subordinate to certain coadjoint Kirillov orbit. A theorem gives the maps \(\theta^j_k:V_k\rightarrow L^2(H\backslash\tilde{G})\) – called the periodizing maps – which generalize the Weil maps [A. Weil, Acta Math. 111, 143–211 (1964; Zbl 0203.03305)]. Given \(\phi\in L^2(H\backslash \tilde{G})\), the periodized image \(\theta^j_k\phi\in\Gamma(M,\ell^{\otimes k})\) lifts to a periodic function \(\vartheta^j_k \phi:G\rightarrow \mathbb{C}\). Then a left-invariant metric on \(G\) is chosen. Associated to the resulting metric on \(M\) there is the Laplacian \(\Delta^{(k)}\) acting on \(\Gamma(\ell^{\oplus k})\), and \(\Delta^{(k)}\) induces a Laplacian \(\Delta_k\) on \(V_k\). Finally, for each \(k\in\mathbb{N}\), there is constructed explicitly a family of maps \(\left\{\theta^{m,n}_k:L^2(\mathbb{R}^2)\rightarrow L^2(M,\ell^{\oplus k}), m,n=0,1,\dots,2k-1 \right\}\) such that \(L^2(M,\ell^{\oplus k}) \simeq\bigoplus_{m,n=0}^{2k-1}\theta^{m,n}_k(L^2(\mathbb{R}^2))\) is an orthogonal decomposition of \(L^2(M,\ell^{\oplus k})\) into irreducible \(\tilde{G}\)-spaces. If \(f:\mathbb{R}^2\rightarrow\mathbb{C}\) is square integrable function, then \(\vartheta^{m,n}_kf\in L^2(\mathbb{R}^4,\mathbb{C})\) verify 4 pseudoperiodicity conditions. It is shown that if \(\psi_0\) is the unique ground state solution of the second-order elliptic operator differential operator \(\Delta_k:=\partial_{xx}-\partial_{tt}+16k^2\pi^2(x^2+t^2) +16k^2\pi^2x^2(x^2/4-t)\), then the images \(\vartheta^{m,n}_k\psi_0\), \(m,n=0,1, \dots, 2k-1\) are \(\vartheta\)-functions associated to the Kodaira-Thurston manifold and form a basis for the almost Kähler quantization of \(M\). The paper is almost selfcontained, including a nice presentation of the classical theory of \(\vartheta\)-functions, geometric quantization [N. Woodhouse, Geometric quantization. Oxford: Oxford Mathematical Monographs (1980: Zbl 0458.58003)] and other theorems used in the construction.