Definitely possible. You want 2 letter extensions like ".hq"? There are many country specific TLDs which are of 2 characters like .in,.us,.fr,.uk and the list goes on. But, before you say your client its possible, ask him "why" they really need this and get to understand their real need
Well it's part of their company name. It's the third word in their name. The .com and .net are taken. How would I go about getting that domain name?
Are you asking about .hq in question or another extension? If I have to answer the thread initial question then, no, you cannot register a domain with just any random 2 letter extension. The TLD's are chosen after there is a proposition, consideration and approval from the official authorities, which is quite a long process. And even then many TLDs soon fail into oblivion due to low demand. Indeed there are a lot of 2 letter country-specific TLDs, however many of them have additional restrictions, often to entities or businesses that reside in that country. So their registration is not always available (and you probably won't need it unless you are going for a local approach on local residents). Back to your example .HQ is not a valid domain extension so you cannot register a domain with it. For a full list of TLDs you can check here: http://en.wikipedia.org/wiki/List_of_Internet_top-level_domains Hope this helps
Tell your customer that using a non stable ccTLD is a high risk and he will not be unable to pay for legal issues or deal with regional problems 1000s miles away from his home. COM have a centralized dispute procedure. I had an experience with a ccTLD that two years to get it back, and as you assume for 2 years it was down.