Fractional jet First Officer reported receiving a GPWS terrain alert on approach to GJT airport; citing fatigue as a factor.

Narrative

The event started with the ILS 11 into GJT. The approach was right to minimums. The Controller vectored us high for the inbound course but in my opinion not too high to correct for. ATC asked if we would like to be re-vectored onto final and I told her that it should be fine. However possibly due to distraction; (it was his last tour at company before retiring) the Captain (PF) did not increase his rate of descent to set us up to reach the FAF at an appropriate altitude. Due to the time between writing this and the event I don't remember specifics as to what I said to him; but I remember saying that we need to increase descent rate now so that we aren't high after we start the approach. At this point in my opinion we are both somewhat distracted. We pass the FAF and the Captain was in VS following the GS and slightly off course but paralleling the LOC. Due to fatigue from this day and the trip in general; I didn't notice until we were later in the approach. ATC asked if we had the field in sight; I said negative. It was at this point; I noticed the course deflection on the CDI and stated it aloud. Shortly after this we had a yellow terrain warning from which I remember telling the Captain to go around. ATC proceeded to give us headings for re-vector and we landed after the second approach. Suggestions: For starters from this event I will definitely look at my own fatigue more conservatively especially when operating into/out of potentially complex airports with terrain; procedures; or weather. Second; when ATC asks if we need a re-vector; I should have asked the Captain what he prefers instead of just responding as if I'm PF. From my prior 121 experience our altitude was nothing out of the ordinary; still I should have asked what he was comfortable with. Third; possibly a conversation between us both before leaving ZZZ would have been beneficial. Following the [internal company] checklist; we may have determined that we should not fly.