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The goal of the present paper is to deliberate certain types of metric suchas $*$-$\eta$-Ricci-Yamabe soliton on $\alpha$-Cosymplectic manifolds withrespect to quarter-symmetric metric connection. Further, we have proved somecurvature properties of $\alpha$-Cosymplectic manifolds admittingquarter-symmetric metric connection. Here, we have shown the characteristics ofthe soliton when the manifold satisfies quarter-symmetric metric connection on$\alpha$-Cosymplectic manifolds. Later, we have acquired Laplace equation from$*$-$\eta$-Ricci-Yamabe soliton equation when the potential vector field $\xi$of the soliton is of gradient type in terms of quarter-symmetric metricconnection. Next, we have developed the nature of the soliton when the vectorfield is conformal killing admitting quarter-symmetric metric connection.Finally, we present an example of a 5-dimensional $\alpha$-cosymplectic metricas a $*$-$\eta$-Ricci-Yamabe soliton with respect to a quarter-symmetric metricconnection to prove our results.

$\ast$-$\eta$-Ricci-Yamabe solitons on $\alpha$-Cosymplectic manifolds with a quarter-symmetric metric connection