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We present the confirmation of a compact galaxy group candidate, CGG-z4, at $z=4.3$ in the COSMOS field. This structure was identified by two spectroscopically confirmed $z=4.3$ $K_s$-dropout galaxies with ALMA $870\rm\, \mu m$ and 3 mm continuum detections, surrounded by an overdensity of NIR-detected galaxies with consistent photometric redshifts of $4.0<z<4.6$. The two ALMA sources, CGG-z4.a and CGG-z4.b, are detected with both CO(4-3) and CO(5-4) lines. [CI](1-0) is detected on CGG-z4.a, and H$_{2}$O($1_{1,0}-1_{0,1}$) absorption is detected on CGG-z4.b. We model an integrated spectral energy distribution by combining the FIR-to-radio photometry of this group and estimate a total star formation rate of $\rm\sim2000\, M_{\odot}$ yr$^{-1}$, making it one of the most star-forming groups known at $z>4$. Their high CO(5-4)/CO(4-3) ratios indicate that the inter-stellar mediums (ISMs) are close to thermalization, suggesting either high gas temperatures, densities, and/or pressure, while the low [CI](1-0)/CO(4-3) line ratios indicate high star formation efficiencies. With [CI]-derived gas masses we found the two galaxies have extremely short gas depletion times of $99$ Myr and $<63$ Myr respectively, suggesting the onset of quenching. With an estimated halo mass of $\rm log (M_{\rm halo}[M_{\odot}])\sim12.8$, we suggest that this structure is likely in the process of forming a massive galaxy cluster.