JSTOR is a very rich source of material on music in general and there is a great deal on Shostakovich and the quartets here. Reviews on each of the quartets and more.
There are so many avenues for finding material at a large university library. But Google Scholar can be extremely helpful. If you search in Russian you will find a large number of articles. There are also very useful Russian webistes such as http://magazines.russ.ru that will provide access to the full text of many of the literary-cultural journals, many with full text for the last 15 years.
There is a website that provides an overview on each of the 15 quartets, Shostakovich: the string quartets. There are also numerous avenues for locating reviews of individual performances: any one of the databases on music, Google Scholar, online newspaper archives, such as the Historical Archive of the New York Times (UIUC database).
There are a wealth of sources available through most libraries that will make finding material on Shostakovich fairly easy. There is one thing to keep in mind. Cyrillic alphabet words are rendered in a variety of ways into latin characters. See "A Few Considerations" for a list of possible spellings.
The number of results you retrieve from a search will be radically affected by the spelling you use. Some of the most useful"subscription" databases include: