At no point are we told that it is Starbucks, and Starbucks has not paid any money for the promotion. We know that he likes his coffee Black, Strong, and possibly Jamaican Blend, but we are not given a brand. The cups look like Starbuick cups, but many coffee companies use similar cups.
I think its some sort of Jamaican blend or house blend. (Mark Harmon said in an interview a while back that most of the time, there is tea in his 'coffee' cups on screen; he only drinks one cup of coffee a day). :-]