  We present the results of $^{69,71}$Ga-NMR measurements on an organic
antiferromagnet $\lambda$-(BEDSe-TTF)$_2$GaCl$_4$
[BEDSe-TTF=bis(ethylenediseleno)tetrathiafulvalene], with comparison to reports
on $\lambda$-(BETS)$_2$GaCl$_4$ [BETS=bis(ethylenedithio)tetraselenafulvalene]
[T. Kobayashi et al., Phys. Rev. B 102, 235131 (2020)]. We found that the
dynamics of two crystallographically independent ethylene groups induce two
types of quadrupolar relaxation in the high-temperature region. As the ethylene
motion freezes, hyperfine (HF) interactions develop between $\pi$ spin and Ga
nuclear spin below 100 K, and thereby magnetic fluctuations of the $\pi$-spin
system are detected even from the Ga site. The HF interaction in
$\lambda$-(BETS)$_2$GaCl$_4$ was more than twice as large as in
$\lambda$-(BEDSe-TTF)$_2$GaCl$_4$, implying that the short contacts between Cl
atoms and the chalcogens of fulvalene part are essential for the transferred HF
interaction. We propose that NMR using nuclei in anion layers is useful for
studying interlayer interactions in organic conductors, which have not been
studied experimentally. In addition, because the mechanism of the transferred
HF interaction is considered to be the same as $\pi$-$d$ interaction in
isostructural Fe-containing $\lambda$-type salts, our findings aid in the
understanding of their physical properties.
