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I have a pole-like shape (octagonal prism) with a bevel added on both its ends and branched into two. The faces are triangulated and the triangles are converted to quads. I applied a subdivision surfaces modifier. The edges of the ends are not perfectly circle, though; there's difference between the bevel thickness with subdivision surfaces.

screenshot 1

screenshot 2

screenshots album

How can I fix this and make the ends look less like squares and more like circles?

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To make subdivided edges look as expected make the corner loops follow the curvature of the desired shape (i.e. circle).

In this case, dissolve quad face in the middle (select edges, press X > Dissolve Edges), inset resulting Ngon twice (press I with face selected), select inner edge loop and merge it at center (Alt+M > At Center).

screenshot of model in progress

Note that you might want to dissolve also 4 edges in the resulting cap to get rid of tris.

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I would delete the four edges that make up the "square" in the middle in Edit Mode. Then select the vertices around the opening that was created. Then press Shift + Alt + S and drag your mouse to create a circle shape.

With that circle opening still selected press E to extrude and S to scale the extrusion down. Then press E again and Right Mouse Click.

Then press ALT + M and select Collapse from the drop down menu to close it.

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