Are the ailerons for a Gulfstream III controlled cables with hydraulic boost? Also, is it possible that an aircraft mechanic could easily access the control to tamper with it?
Yes, the flight control surfaces on the Gulf Stream III are moved by cables with hydraulic assistance.
From a study guide for the type.
The flight controls on the G III are cable driven with hydraulic boost.. They are powered by both the "Combined" and "Flight" hydraulic systems. Each of the systems provides 1500 psi to the flight control servo actuators. When the landing gear or flaps are extended, the combined system pressure increases to 3,000 psi. If one system fails, the remaining system pressure increases to 3000 psi. Failure of one hydraulic system therefore, does not result in a loss of control effectiveness.
As to whether a mechanic could interfere with the controls, of course. But for what purpose? I am not sure that I understand what you mean by "tamper" but I assume that you mean sabotage. If so, then there are multiple ways to do this but I see no point in speculating as to what someone might do or why.