An interaction term in the Lagrangian which is trilinear leads to a 3particle vertex, while one which is quadrilinear leads to a 4particle vertex. These things occur for example in electroweak theory. The kinetic energy term for the W boson is 1/4 W_{μν}W^{μν} where W_{μν} = ∂_{μ}W_{ν}  ∂_{ν}W_{μ}  gW_{μ} x W_{ν}, which leads among other things to a 4particle WWWW vertex. Also there is a WWhh vertex from its interaction with the Higgs.
