The ornamental design for a video conference system enclosure, as shown and described.
FIG. 1 is a perspective view of a video conference system enclosure, showing novel design, the view showing the right side, the left side is identical to the right side; and,
FIG. 2 is a right side view of the video conference system enclosure of FIG. 1.
The broken lines in the drawing form no part of the claimed design. The bottom of the video conference system enclosure is flat and unornamented.