Avoid division by zero in PostgreSQL

You can use NULLIF function e.g.

something/NULLIF(column_name,0)

If the value of column_name is 0 - result of entire expression will be NULL

Since count() never returns NULL (unlike other aggregate functions), you only have to catch the 0 case (which is the only problematic case anyway). So, your query simplified:

CASE count(column_name)
   WHEN 0 THEN 1
   ELSE count(column_name)
END

Or simpler, yet, with NULLIF(), like Yuriy provided.

Quoting the manual about aggregate functions:

It should be noted that except for count, these functions return a null value when no rows are selected.