Through this paper we analyze from first-principles, high-precision derivation of the spectral shape, characteristic amplitude, and unique observational signatures of the stochastic gravitational wave background (SGWB) generated during the primordial first-order topological phase transition that is a fundamental prediction of the Expanded Quantum String Theory with Gluonic Plasma (EQST-GP) framework. The transition corresponds to the spontaneous symmetry breaking \( SU(4) \to SU(3)_C \times U(1)_{\text{DM}} \) within the gluonic plasma confined to M5-brane world-volumes in the specific compactification geometry \( M_4 \times \text{CY}_3 \times S^1/\mathbb{Z}_2 \) with Euler characteristic \( \chi(\text{CY}_3) \approx -960 \). We move beyond generic parameterizations to perform a complete microphysical calculation. Starting from the finite-temperature effective potential for the symmetry-breaking scalar field \( \Phi \), where the coefficients \( D, T_0, E, \lambda \) in \( V_{\text{eff}}(\Phi, T) \approx D (T^2 - T_0^2) \Phi^2 - E T \Phi^3 + (\lambda/4) \Phi^4 \) are not free parameters but are explicitly computed from the underlying M-theory parameters: the M5-brane tension \( T_{M5} = (2\pi)^{-5} l_P^{-6} \), the volumes of the wrapped 2-cycles \( \text{Vol}(\Sigma_2) \), the stabilized values of the Kähler moduli from the KKLT-inspired potential \( V_{\text{up}}(\phi) \), and the thermal contributions of the confined \( SU(4) \) gluon degrees of freedom and the associated moduli fields. This derivation yields a highly specific set of phase transition parameters: a critical temperature \( T_c = 1.04^{+0.06}_{-0.05} \times 10^{16} \, \text{GeV} \), a nucleation temperature \( T_n = 0.971 \times 10^{16} \, \text{GeV} \) (corresponding to a Euclidean action \( S_3(T_n)/T_n = 138.2 \)), a transition strength parameter \( \alpha = 0.42 \pm 0.03 \) defined as the ratio of latent heat density to radiation energy density \( \alpha = \epsilon / \rho_{\text{rad}} \), and an inverse transition duration relative to Hubble \( \beta / H_* = 94.7 \). The bubble wall velocity \( v_w \), determined from the balance of the vacuum driving pressure against the friction from the strongly-coupled (2,0)-theory plasma on the M5-branes, is calculated to be \( v_w = 0.27 \, c \), characteristic of a deflagration mode. We then compute the gravitational wave spectrum \( \Omega_{\text{GW}}(f) h^2 \) from the three principal sources—scalar field bubble collisions \( (\Omega_\phi) \), sound waves in the post-collision plasma \( (\Omega_{\text{sw}}) \), and magnetohydrodynamic turbulence \( (\Omega_{\text{turb}}) \)—using the most advanced hydrodynamic simulations and envelope approximations, adapted for the specific relativistic degrees of freedom \( g_* = 187 \) of the EQST-GP plasma. The total spectrum exhibits a distinct, multi-peak fingerprint: a primary peak from sound waves at \( f_{\text{sw}} = 1.87 \times 10^{-3} \, \text{Hz} \) with amplitude \( \Omega_{\text{GW, sw}} h^2 = 6.31 \times 10^{-14} \), a secondary, broader peak from turbulence at \( f_{\text{turb}} \approx 3.2 \times 10^{-3} \, \text{Hz} \) with \( \Omega_{\text{GW, turb}} h^2 \approx 1.2 \times 10^{-14} \), and a high-frequency tail from bubble collisions. Crucially, we establish a detailed discrimination strategy demonstrating that the EQST-GP signal is distinguishable from inflationary tensor modes, cosmic string networks, and generic first-order phase transitions through multi-messenger consistency with predictions for ultra-heavy Majorana gluon dark matter, Hubble tension resolution, and fundamental constant derivation. We present a comprehensive detection blueprint for LISA, demonstrating that a signal-to-noise ratio is achievable over a 4-year mission with optimal template-based analysis, and outline how cross-correlation with future CMB B-mode polarization measurements and 21-cm cosmology observations can further isolate this signal from astrophysical foregrounds.