What is the mechanism of oxidation of phenol to benzoquinone?
The reaction is used industrially to make hydroquinone (source: wikipedia), but often using hydrogen peroxide as the oxidation agent in the presence of a catalyst.
A study on catalytic hydroxylation of benzene comes to the following conclusion:

experimental evidences have allowed first discarding Fenton-like mechanisms and later proposing the plausible competition between an electrophilic aromatic substitution pathway with the alternative rebound (hydrogen abstraction) route.

The Fenton-like mechanism refers to addition of a free oxygen radical species to benzene, which they reject based on experiments with radical scavengers. Instead, they hypothesize formation of a radical oxygen:catalyst complex. This either reacts with benzene in an electrophilic substitution, or abstracts a hydrogen from benzene, which then rebounds and reacts with the oxygen still attached to the catalyst.
Although the study I am citing is for hydroxylation of benzene instead of phenol, the paper also talks about substituted benzenes.

[OP:] My guess is that the double bond of phenol will react with the oxidizing agent, but not sure of how this will work.

You can't pin down a single double-bond in an aromatic system. What you can say is that the existing hydroxyl group will steer the reaction to make an ortho- or para- product. Once hydroquinone is formed, it oxidizes quickly to benzoquinone under the oxidizing conditions.

[OP:] What is the mechanism of oxidation of phenol to benzoquinone?

For the reactions conditions given by the OP, I did not manage to find a mechanism. However, the mechanisms I discussed probably are sufficiently general that they apply in this special case as well. That is just a guess, though.