Trademark an LLC? Sure. Why not? Go for it.
There's nothing about a business being an LLC (limited liability corporation) as opposed to, say, a partnership or sole proprietorship, that would prevent it from registering a trademark. But keep in mind that trademarks mostly protect individual products and services, rather than company names. An LLC has to work within the existing trademark system, just like any other business.
Need more information? Newspaper archives are one of the best online research tools.