The Chandos, as recommended by EcclesCake, is one of the nearest pubs to Trafalgar Square. As has been said, the upstairs bar is relaxing, and the beer is cheap. The downside (for some people) is that it's a Samuel Smith's pub, which only sells Sam Smith's products. (So you can't buy draught Guinness or a well-known brand of bitter or lager. You have to buy Sam Smith's stout, bitter or lager). Some of their drinks are OK but others are only 'second rate'.
If you're going to try a Sam Smith's pub, the Lyceum Tavern (on the Strand, near the Aldwych, just on the edge of Covent Garden) would be my own preference. (You might well see me in there!). It's got secluded little alcoves downstairs and big, comfy sofas upstairs. (The draught wheat beer, which is one of Sam Smith's better products, is highly recommended by me).
Although it's outside of your area, you might like to know about the Betjeman Arms, at St Pancras International station. It's wonderfully relaxing, with deep, comfy sofas, good real ales and excellent food.
Chris