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A novel fully quadrature subcarrier-index shift keying orthogonal frequency division multiplexing (FQSISK-OFDM) modulation scheme for underwater acoustic communications (UWAC) is proposed in this paper. In the FQSISK, the data is conveyed by active and non-active OFDM-subcarriers indices. Moreover, the active odd and even subcarriers are marked with separable predefined dummy variables (DV). Herein, the DV <inline-formula> <tex-math notation="LaTeX">$\pm \delta $ </tex-math></inline-formula> (i.e., it can take any predefined value), and its quadrature <inline-formula> <tex-math notation="LaTeX">$\pm j\delta $ </tex-math></inline-formula> are exploited to distinguish between active even and odd subcarriers. Hence, the possibility of power leakage from active to non-active subcarriers is reduced, which minimizes the effect of severe UWAC inter-carrier-interface (ICI). Moreover, the spectral efficiency of the FQSISK hardly depends on the subcarriers grouping size, which reduces the proposed FQSISK decoder complexity significantly. The simulation results demonstrate the proposed FQSISK superiority, as it achieves higher spectral and energy efficiencies than the plain OFDM and the recently introduced OFDM index modulation (OFDM-IM) techniques.