The actual physical turning of the screw jack that moves the THS is performed by two hydraulic motors, one powered by the green system, the other by the yellow. The trim wheel is a control only - it cannot turn the screwjack itself. So you need green or yellow hydraulics, (usually both). So you need at least one engine running.
I suppose that in theory you could move the THS with the yellow electric pump pressurising the yellow system, but I never tried it and don’t know if there is a system restriction against this.