Generalising Omar's answer, braided monoidal categories are tricategories with one 1-morphism. Symmetric monoidal categories are tetracategories (whatever that is) with one 2-morphism. Writing these correspondences out explicitly, you should be able to work out the $\omega$-nerve of the underlying $\omega$-category (Ross Street's work with orientals). Actually, following Street's approach, an $\omega$-category already is a simplicial set and one would expect that you can expect a tricategory to be just an $\omega$-category where all the $n$-morphisms are identities above 3.
I believe the Duskin nerve is just taking a bicategory as an $\omega$-category following Street's approach and then considering its nerve.