I've looked at some websites where employees of Marriott talk about working for the company, and it looks that, yes, Marriott does conduct drugs tests.
Marriott will conduct a urine test after the interview stage of your job application process and usually will have the results ready in around three days.
I have read of Marriott using swab tests but none of these were within the last five years or so, so I guess they've given up swab tests for urine tests.
If you're going for an interview at Marriott, I hope it goes well. Here, a former Marriott employee talks about applying for a job at the company: