Gulfstream G650 pilot reported a false localizer capture resulted in a course deviation and a terrain warning from ATC.

Narrative

We were positioning from ZZZZ to ZZZZ1 after an overnight flight from ZZZ. The flight was going smoothly; and the Approach Controller asked if we had passengers onboard. We informed him that we did not have passengers; and he asked if we wanted a steeper descent that would allow for a quicker arrival. We had been flying for a significant time and choose the steeper descent. As we got closer to the airport the Controller cleared us to ZZZZZ (IF). Our intercept angle was approaching 90 degrees; but we were visual and didn't consider this to be an issue. About 4 miles from ZZZZZ the localizer captured and started a turn toward the airport. We recognized fairly quickly that it was a false capture and the PF disconnected the auto pilot. At that time the Controller directed us to make an immediate turn to heading 360 for terrain. We started the turn and told I him that we had the terrain in sight and he asked if it had been a false capture to which replied that it had been. We continued our turn to 360 degrees and he asked if we could accept direct to ZZZZZ and continue the approach. We accepted the clearance and had no other issues with the flight and landing. I believe that with the long day we were tired and would have recognized the false capture earlier had we not been on extended duty. Although I don't believe we were ever in any hazardous position due to the weather being VFR.