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Twisting moiré heterostructures to the flatband regime allows for the formation of strongly correlated quantum states, since the dramatic reduction of the bandwidth can cause the residual electronic interactions to set the principal energy scale. An effective description for such correlated moiré heterostructures, derived in the strong-coupling limit at integer filling, generically leads to spin-valley Heisenberg models. Here we explore the emergence and stability of spin-liquid behavior in an SU(2)^{spin}⊗SU(2)^{valley} Heisenberg model upon inclusion of Hund's-induced and longer-ranged exchange couplings, employing a pseudofermion functional renormalization group approach. We consider two lattice geometries, triangular and honeycomb (relevant to different moiré heterostructures), and find, for both cases, an extended parameter regime surrounding the SU(4) symmetric point where no long-range order occurs, indicating a stable realm of quantum spin-liquid behavior. For large Hund's coupling, we identify the adjacent magnetic orders, with both antiferromagnetic and ferromagnetic ground states emerging in the separate spin and valley degrees of freedom. For both lattice geometries the inclusion of longer-ranged exchange couplings is found to have both stabilizing and destabilizing effects on the spin-liquid regime depending on the sign of the additional couplings.