I'm trying to do the following query
INSERT INTO 'table_1'(column_1)
SELECT DISTINCT 'column 2'
The way you've written your query currently is that you're trying to
INSERT a string into your integer column. When you do
SELECT DISTINCT 'column2'
you are selecting the literal string
'column2' and not the values from that column. I would unquote the references to your table and column:
INSERT INTO table
SELECT DISTINCT column2 FROM ...