Q:"Can a buisness trademark a name being used only for Facebook username"
A: YES. You misuse the term "trademark", which is a noun not a verb. Trademark protection arises AUTOMATICALLY upon use, so yes if there is use there is trademark protection. That is hardly the end of the story, however.
Q:"they are using the name for there[sic] facebook username but it is not there[sic] intended buisness name by any means and not advertised other then[sic] fb username. "
A: So what. No law requires using the same trademark [and you probably mean servicemark rather than trademark] everywhere. Businesses often use lots of different brand names for various reasons, usually better marketing, such as to differentiate a premium line from an economy line. GMC has Chevrolets and Pontiacs and Cadillacs and each of those does not normally mention GMC unless they are selling trucks. That does not make any one of them less a brand name.
If you have to ask this question, you need to use a trademark lawyer to get things right and protect things in the proper way and to steer clear of infringement of trademarks you apparently hope are not trademarks.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
First, the name of a business [as opposed to the name of a product or service] is a TRADE NAME. That trade name may, under some circumstances, also serve as a TRADEMARK -- but only if it's also actually used as a trademark.
Second, a business must ALWAYS have at least one trade name and it may have more. The primary trade name is the one registered with whatever governmental regulating body regulates the business [town or county Business License Office or the state's Secretary of State or whoever]. Any other trade name is a "fictitious business name." Which must, in most states, also be registered.
So, yes, a business may do business offline under one name and online under another.
Are either business names in your situation also trademarks? Dunno. Only your own trademark attorney, licensed to practice in your state, can help you answer that question.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
If you want to know whether any given word/phrase can be registered for a trademark, you need to hire a professional to determine the viability and availability of that proposed trademark.
Facebook is only one of the places a professional would check, and you need to see EVERY use of the same or confusingly similar TM for the same or confusingly similar goods or services that you want to offer, not just this one FB use that you know of.
See a trademark lawyer for help. About 1/2 of all trademark applications get rejected, and the fees are non-refundable, not to mention the time and energy that could be wasted if the choice is a bad one
Avvo doesn't pay us for these responses, and I'm not your lawyer just because I answer this question or respond to any follow-up comments. If you want to hire me, please contact me. Otherwise, please don't expect a further response. We need an actual written agreement to form an attorney-client relationship. I'm only licensed in CA and you shouldn't rely on this answer, since each state has different laws, each situation is fact specific, and it's impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.