Fairbrass, who joined from brewer SABMiller in 2015, is currently on gardening leave until early next year, having stepped away from an active role last month.
His position has disappeared in a cost-cutting move. It will be replaced with a head of global media, with G4S currently recruiting for that role.
Meanwhile, Baker has resigned from the company, which he joined in 2012, PRWeek understands.
A source linked their departures to disagreements about the direction of the firm, which has undergone a challenging few years. G4S has recently been accused of dragging its feet on restructuring plans, although earlier this month it reported an almost doubling of half-year, pre-tax profit to £218m on revenue of £3.7bn.
G4S faced severe criticism in the months before and following the London 2012 Olympics, when it failed to provide enough security personnel to fulfil its £284m contract.
The source told PRWeek: "G4S has been a tough place to work in the past few years, and 2016 saw it hit by crisis after crisis. There has clearly been some difference of opinion on the scale of investment required to address the company's reputation challenges, and these departures signal a scaling back of ambitions."
However, group corporate affairs director Debbie Walker told PRWeek there had been "no disagreement, change of strategy or change in the level of investment", adding that "teams evolve over a period of time". She said G4S did not comment on specific colleagues.
Fairbrass and Baker were part of a four-strong global media team at G4S.
Fairbrass stepped down from SABMiller in 2014, having spent more than ten years at the company, joining in 2003 as investor relations manager and becoming head of media relations a year later. He was SVP of global comms for three years before he left.
Fairbrass previously worked at financial PR firms Brunswick Group and Tulchan Communications.
Baker first joined in G4S as a public relations executive before working as press and projects officer at the College of Policing. He returned to the security company in 2014, first as media manager before becoming head of UK media.