Yes, according to numerous websites and job advertisements, including this one that was posted on LinkedIn by the company itself, Expedia does drug test:
According to the company, sales agents will need to pass a pre-employment drug screening test, as well as a background check as a part of their initial assessment.
Individuals who fail to pass this section of the application will not be considered for a position within the company.
Once this pre-employment screening has been administered, the necessity for further tests will depend on the contract between the employee and the company.
It is believed that most employees will not have to undergo further testing, either random or otherwise, although this is not set in stone.
The drug testing conducted by Expedia is limited to urine testing though, so more accurate (and more expensive) methods such as hair testing aren't required.
If you've recently taken an illegal substance, then I'm sure this page on how to pass urine drug screens will help: //drug-alcohol-testing.blurtit.com/205590/how-can-i-pass-the-urine-test