Citation Excel First Officer reported flight control issues caused controllability problems in flight. The Flight Crew elected to make a precautionary landing at destination airport and requested maintenance.

Narrative

This Part 135 passenger leg was the first leg that I flew in this particular aircraft. (The Pilot In Command and I had flown another aircraft; another tail number; for the proceeding five days of our eight day rotation.) When we received this aircraft out of maintenance; the Pilot In Command flew the first leg in this new tail number; and he commented that the elevator trim indicator (on the trim wheel) did not reflect the actual position of our elevator trim. He noted that the elevator controls did not 'feel right' on his leg; and thus; he visually inspected the actual trim location against the trim indicator position. (It is noted that the Pilot In Command is also an A&P; and he chose not to write up the issue; as he said it was 'still within limits.') It was my leg from ZZZ to ZZZ1; and I was flying from the right seat. We utilized a flap setting of 15 degrees for the takeoff; and everything seemed to feel 'normal' through the take-off roll; rotation; and climbing through 500 ft. I utilized the electric trim on my yoke; and it was uneventful until the autopilot was engaged. (It is further noted that this particular Pilot In Command was challenging to fly with; as he clearly felt that he knew much more than me and 'knew better' in general. It was also clear that he preferred to utilize certain aircraft automation on the initial climb out i.e.; 'flight level change' while I preferred to utilize 'vertical speed' on the initial climb out; yet he would engage 'flight level change' on my legs nonetheless.) Before engaging the autopilot; I set the trim appropriately for the climb out. Once the autopilot was engaged; I felt the yoke push forward. It seemed very nose-heavy and the plane descended a couple of hundred ft. The Pilot In Command said 'climb; climb'. However; the extremely nose heavy condition made the climb difficult. (It was so startling that this occurred; even though the plane was perfectly trimmed prior to autopilot engagement; that I was uncertain whether the Pilot In Command had engaged any other automation; like 'flight level change'; that could have caused this downward pitch to occur.) The alert system announced 'sink rate;' and the Pilot In Command took controls. The Pilot In Command struggled to climb the aircraft as well; and the alert system continued to announce 'sink rate.' After several seconds; the climb rate gradually improved. Our one passenger inquired about this event; as our passenger thought we were returning to the airport. (The Pilot In Command told the passenger that everything was fine; but that I was not prepared for the amount of nose-heaviness in this aircraft!) The Pilot In Command did not want to discuss this event with me; and he refused to discuss the possibility that something was mechanically deficient with the plane. The rest of the trip from ZZZ to ZZZ1 was uneventful; however; I was extremely concerned for the remainder of this rotation; as I felt that the trim rigging needed to be checked. I did not fly any additional legs in this aircraft during the remainder of the rotation.