  Twisting moir\'e 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 moir\'e
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)$^{\mathrm{spin}}\otimes$SU(2)$^{\mathrm{valley}}$ 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 moir\'e
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.
