In PSQL, is it possible to insert rows in two different tables with not-nullable foreign key columns referencing each other?
Assuming the primary key columns are defined as serial (or bigserial) you can first get the new IDs using nextval() and then use them in the subsequent inserts. 

This can either be done separate statements:

select nextval(pg_get_serial_sequence('make','make_id'));
select nextval(pg_get_serial_sequence('model','model_id'));

insert into make 
  (make_id, name, first_model_id) 
values (
  currval(nextval(pg_get_serial_sequence('make','make_id'))), 
  'Make One', 
  currval(pg_get_serial_sequence('model','model_id'))
);

insert into model
  (model_id, name, make_id)
values (
  currval(pg_get_serial_sequence('model','model_id')),
  'Make One', 
  currval(nextval(pg_get_serial_sequence('make','make_id')))
);


or a single statement using a data modifying CTE

with new_ids as (
   select nextval(pg_get_serial_sequence('make','make_id')) as new_make_id, 
          nextval(pg_get_serial_sequence('model','model_id')) as new_model_id 
), new_make as (
  insert into make (make_id, name, first_model_id) 
  select new_make_id, 'Make One', new_model_id
  from new_ids
)
insert into model (model_id, name, make_id) 
  select new_model_id, 'Model 1', new_make_id
from new_ids;