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);

table.column1 can be confused with a schema lookup. It thinks that it needs to look for a schema named table.

Be sure to check out the docs on how to create a schema. It says, that

[..] the even more general syntax

database.schema.table

This would also make no syntactical sense since you already know what table you are referring to. IF you need to query the schema, you would write it.

INSERT INTO your_schema.table (column1, column2) VALUES (1, 2);