My test source data is a file called "poly_town", almost the same as in your picture, i.e. approximately a grid.

- Extract vertices with the geo tool:
Vector>Processing geometry>Extract vertices, see the figure below for the result.

- Create combined buffers on the selected nodes using the geo tool:
Buffer as shown in the figure below

See the figure below for the expected result.

- Extract centroids on the combined buffer zones at the intersections of blocks, using the geoinstrument:
Vector>Processing of geometry>Centroids, the result see the figure below

- Perform Delaunay triangulation on these centroids, using:
Vector>Processing geometry>Triangulation Delaunay, hereinafter
convert polygons into lines with the geo tool: Vector>Processing geometry>Convert polygons into lines,
then blow them up with: Analysis Tools>Vector Geometry>Blow up Lines, see the figure below

Select the required blasted lines that do not fall into quarters using the geo tool: Analysis tools>Vector selection>Extract by spatial position see the figure below

Delete unnecessary artifacts and save the result as a new file using <Export>-<Save objects as... >...
see the figure below
