Two natural centers of mass of a quadrilateral are not usually at the same point.
Suppose equal masses are placed at the vertices of the quadrilateral. We can pair up the vertices into two pairs. The center of mass of each pair is the midpoint of the segment connecting them. So the center of mass of the 4 points is the midpoint of the segment connecting the two midpoints. It is also the intersection of two segments that we can obtain by connecting midpoints of pairs, when we group the 4 points into pairs two different ways.
We see this in the figure. The center of mass is on either segment connecting the midpoints of opposite sides.
We think of the quadrilateral as being made of some thin, flat material. We want to find the center of mass of this object. We will call it the center of mass of the region, or the area center of mass. It is known that for a triangle the area center of mass is the same as the vertex center of mass, which is the centroid. The centroid is the point where the medians come together.
Thus if we break the quadrilateral into two triangles and find the centroid of each, the center of mass of the quadrilateral is on the segment connecting the two centroids. If we break up the quadrilateral two different ways, we get two such segments. The intersection is the center of mass of the region.