How to Protect one -OH group in 1,4-benzenedimethanol using TBDMS
The product is symmetrical so it doesn't really matter which alcohol gets protected, you'll end up with the same product. 

There are many examples of doing this kind of mono protection of simple substrates. Quite often a fairly large excess of the diol is added to try and ensure no bis protection takes place. In your case the starting material should be easily separable from product by column chromatography.  

A potentially better method is to formally deprotonate the diol with 1 eq of a base such as sodium hydride or butyl lithium then trapping with a solution of the silyl chloride. This often works well in cases where bis protection may occur, as energetically its unfavourable for both oxygens on the molecule where there's molecules that are neutral