All you have to do is to define what gate would correspond to each arity hyperedge, if it's just Z, CZ, CZZ, etc., then something like this works (assuming a simple hypergraph with only unique vertices in every hyperedge):

QuantumCircuitOperator[
    Switch[Length[#],
        0,Nothing,
        1,QuantumOperator["1"]->#,
        _,{"C",QuantumOperator["1"]->First[#],Rest[#]}
    ]&/@DeleteDuplicates/@ResourceFunction["RandomHypergraph"][{10,{{3,1},{5,2},{4,3}}}]
]

N.B. In WQF operator "1" is the same as Z, and "0" is -Z

Attachments:

HypergraphCircuit.png