Ulam's Spiral with Oppermann's Diagonals (quarter-squares)

Starting with Graph[u] you can extract the coordinates of the vertices as follows

gr = Graph[u];
crds = AbsoluteOptions[gr, VertexCoordinates];

The graph including the diagonals can then be drawn according to

Graph[VertexList[gr], Union[u, q1, q2, q3, q4], crds]

Mathematica graphics

You can also use GraphEmbedding to get the vertex coordinates:

Graph[VertexList[g = Graph[u]], Union[u, q1, q2, q3, q4],
      VertexCoordinates -> GraphEmbedding[g]]

enter image description here

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