Your query does not give the expected result, because you only test for 1 of the 3 (or 4) possible ways of overlapping/intersecting:

existing date periods:  +--------------+      +-----+       +----+   +-----+
                            contains        contained by        touches
testing periods:            +------+       +------------+     +---------+

To test every possible way of overlapping, you can use daterange's overlaps operator: &&

select count(*)
from   booking
where  daterange(from_date, to_date, '[]') && daterange('2017-04-20', '2017-04-25', '[]');

But be warned: only using these kind of manual checking will not prevent inserting overlapping date ranges in high concurrency. There is a (small) window, after this test & before the actual insertion for another, concurrent statement to insert a conflicting row. To also avoid that, you can use exclusion constraints:

alter table booking
  add constraint exclude_overlapping_bookings
  exclude using gist (daterange(from_date, to_date, '[]') with &&);

Note: daterange's overlaps operator (&&) works the same way as the overlaps operator, mentioned by @a_horse_with_no_name. Except:

• it supports exclusion constraints
• it supports use of indexes (if you added the exclusion constraint, the index is created automatically for you)
• you can fine-tune how to include the range's boundaries. The default is [), meaning: the lower bound is inclusive, but the upper bound is exclusive. With overlaps, this is the only supported inclusion.
• overlaps works with timestamp with time zone values. Your date values will be casted to that, using midnight as time & the current TimeZone setting. This is may or may not what you want.

http://rextester.com/NNWD52481

Just to examine another way, another query that doesn't use overlap operator, but only AND et OR: (using data posted by a_horse...). This is the logic:

TO_DATE>= [your start date] AND TO_DATE<= [your end date]
OR
FROM_DATE <= [your end date] AND FROM_DATE>= [your start date]

select *
from bookings
where to_date>=date '2017-04-18' AND to_date<=date '2017-04-19'
OR from_date<= date '2017-04-19' AND from_date>=date '2017-04-18';

select *
from bookings
where to_date>=date '2017-04-20' AND to_date<=date '2017-04-25'
OR from_date<= date '2017-04-25' AND from_date>=date '2017-04-20';

Use the overlaps operator:

select count(*) = 0 as allow_booking
from "table"
where (from_date, to_date) overlaps (date '2017-04-20', date '2017-04-25');

That will also take care of bookings that do not completely fall in the range to test.

The query will return true if the booking is allowed, and false if not.

Examples:

create table bookings 
(
  id serial,
  from_date date, 
  to_date date
);

insert into bookings 
  (from_date, to_date) 
values 
  (date '2017-04-20', date '2017-04-20'),
  (date '2017-04-20', date '2017-04-24'),
  (date '2017-04-26', date '2017-04-29');

Then the following

select *
from bookings
where (from_date, to_date) overlaps (date '2017-04-18', date '2017-04-19');

will return nothing, so, count(*) = 0 returns true

The following query:

select *
from bookings
where (from_date, to_date) overlaps (date '2017-04-20', date '2017-04-25');

returns:

id | from_date  | to_date   
---+------------+-----------
 1 | 2017-04-20 | 2017-04-20
 2 | 2017-04-20 | 2017-04-24

So count(*) = 0 returns false

And the query:

select *
from bookings
where (from_date, to_date) overlaps (date '2017-04-27', date '2017-04-28');

will return:

id | from_date  | to_date   
---+------------+-----------
 3 | 2017-04-26 | 2017-04-29

And for that count(*) = 0 is also false.