We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space $\mathbb{QU}$. As a consequence, we show that Isom ($\mathbb{QU}$) has property (T). We also derive several ergodic theoretic consequences from this classification: (i) every probability measure-preserving action of Isom( $\mathbb{QU}$) is either essentially free or essentially transitive, (ii) every ergodic Isom( $\mathbb{QU}$)-invariant probability measure on $[0, 1] ^\mathbb{QU}$ is a product measure. We obtain the same results for isometry groups of variations of $\mathbb{QU}$, such as the rational Urysohn sphere $\mathbb{QU}_1$ , the integral Urysohn space $\mathbb{ZU}$, etc.