  We select and characterise a sample of massive
(log(M$_{*}/$M$_{\odot})>10.6$) quiescent galaxies (QGs) at $3<z<5$ in the
latest COSMOS2020 catalogue. QGs are selected using a new rest-frame colour
selection method, based on their probability of belonging to the quiescent
group defined by a Gaussian Mixture Model (GMM) trained on rest-frame colours
($NUV-U, U-V, V-J$) of similarly massive galaxies at $2<z<3$. We calculate the
quiescent probability threshold above which a galaxy is classified as quiescent
using simulated galaxies from the SHARK semi-analytical model. We find that at
$z\geq3$ in SHARK, the GMM/$NUVU-VJ$ method out-performs classical rest-frame
$UVJ$ selection and is a viable alternative. We select galaxies as quiescent
based on their probability in COSMOS2020 at $3<z<5$, and compare the selected
sample to both $UVJ$ and $NUVrJ$ selected samples. We find that although the
new selection matches $UVJ$ and $NUVrJ$ in number, the overlap between colour
selections is only $\sim50-80\%$, implying that rest-frame colour commonly used
at lower redshifts selections cannot be equivalently used at $z>3$. We compute
median rest-frame SEDs for our sample and find the median quiescent galaxy at
$3<z<5$ has a strong Balmer/4000 Angstrom break, and residual $NUV$ flux
indicating recent quenching. We find the number densities of the entire
quiescent population (including post-starbursts) more than doubles from
$3.5\pm2.2\times10^{-6}$ Mpc$^{-3}$ at $4<z<5$ to $1.4\pm0.4\times10^{-5}$
Mpc$^{-3}$ at $3<z<4$, confirming that the onset of massive galaxy quenching
occurs as early as $3<z<5$.
