We study quasi-stationarity for a large class of Kolmogorov diffusions, that is convergence to equilibrium conditioned to non-extinction. These diffusions arise from population dynamics and are obtained from generalized (as for example logistic) Feller diffusions. We firstly study in details the one-dimensional case. The main novelty is that the drift term may explode at the origin and the diffusion may have an entrance boundary in infinity. We obtain conditions on the drift for the existence of quasi-stationary distributions, as well as rate of convergence, and existence of the process conditioned to be never extinct. We show that under these conditions, there is exactly one conditional limiting distribution (which implies uniqueness of the quasi-stationary distribution) if and only if the process comes down from infinity. The proofs are based on spectral theory. Next, we show that our tools allow us to consider an appropriate multi-dimensional framework related to interesting examples from population dynamics. The uniqueness is then related to the ultracontractivity of the semigroup of the killed process.