No. A person employed on a full-time basis by the Commonwealth (e.g. an Australian Public Service employee or a public office holder) may not receive remuneration for holding a part-time public office but can receive travelling allowance for official travel associated with fulfilling the requirements of that office.
The legislative basis for this is sub-section 7(11) of the Remuneration Tribunal Act 1973 (the Act). The restriction also applies to a person holding any office or appointment or who is otherwise employed, on a full-time basis, in the service or employment of the Commonwealth, the Administration of a Territory, a public statutory corporation, an incorporated company referred to in paragraph 3(4)(da) of the Act or an incorporated company all the stock or share in the capital or which is or are beneficially owned by the Commonwealth or by a public statutory corporation.
Whether a State public servant can receive payment for holding a part-time public office (e.g. on a Federal Government body) is a matter for the relevant State government.