Not quite. But a recent study showed they can recognize four-letter words, and discriminate words from non-words, even when presented with words they've never seen before. This suggests they are able to develop a rudimentary grasp of the rules of word formation.
This is not quite reading because there is no association with sounds or comprehension of meaning. But it may point to one of the fundamental building blocks of human language.
It likely has more to do with object recognition than linguistic abilities.
Baboons and 4-letter words point to origins of reading New Scientist
Baboons recognize words on a screen BBC News