Table-name.column-name in PostgreSQL?
Firstly, if you really want to use the name table for your table then you will need to enclose it in quotes:

create table table (column1 integer, column2 integer);
ERROR:  syntax error at or near "table"

create table "table" (column1 integer, column2 integer);
CREATE TABLE


but assuming your table name is something different (which I highly recommended), the error returned indicates that the parser is interpreting my_table as a column name:

create table my_table (column1 integer, column2 integer);
CREATE TABLE

insert into my_table (my_table.column1, my_table.column2) values (1, 2);
ERROR:  column "my_table" of relation "my_table" does not exist


This interpretation is sensible, as column1 may be a composite type, for example:

create type my_type as (column1 integer, column2 integer);
create table my_table (col my_type);
insert into my_table (col.column1, col.column2) values (1, 2);
select * from my_table;

  col
-------
 (1,2)
(1 row)


You probably just want the following as already suggested by DrColossos:

insert into table (column1, column2) values (1, 2);