A Guardian article on a lawsuit relating to the Klingon language has a photo of Spock. Obviously, he isn't a Klingon, but can he speak the language?
A search of Memory Alpha has no mention of it.
As Broklynite mentions in the comments, the reason a picture of Spock was used is likely because he is the most recognizable symbol of Star Trek, and that picture in particular.
As for whether or not Spock speaks Klingon:
At some point after the events of Star Trek V: The Really Really Bad One, Spock left the Enterprise to become a diplomat, and one of his most important contributions was serving as a special envoy to the Klingon Empire, which would eventually result in the Khitomer Accords.
It would not be too surprising if Spock learned some Klingon during this time, but there is reason to believe that he was far from fluent:
In Star Trek VI: The Undiscovered Country, there is a scene where the Enterprise crew - under Spock's command - is sneaking through the Klingon Empire, bound for Rura Penthe. They come across a Klingon listening post, and have to pass as Klingons. They can't use the universal translator because it would be recognized, so instead they must resort to flipping through dictionaries.
If Spock were fluent in Klingon, we would expect him to be handling the transmission, but instead this is left to Uhura. In fact, if Spock is even decent at Klingon, we'd expect him to be present and helping out with the translation, but instead he is completely absent from this scene.
Spock lived for more an additional 100 years in this universe between these events and the Hobus incident, so it's not impossible that he learned Klingon in that time. However, most of what we know from canon about his life after Star Trek VI comes from the two-part TNG episode Unification and the movies Star Trek XI and Star Trek Into Darkness, none of which involve any contact between Spock and any Klingon.
According to the original script for Star Trek VI: The Undiscovered Country, Spock can't speak Klingon.
If he did, then he would have a) noticed that Uhura's Klingon was bizarrely formal and b) noticed that the Klingons were replying in a bizarrely formal dialect.
VOICE FILTERED (subtitled KLINGON): We art thy freighter... URSVA six weeks out of Kronos... Over...
KLINGON: (subtitled KLINGON) Whither are you bound? Over...
[Another LONG pause. They squint at the delay.]
122 INT. ENTERPRISE BRIDGE
[Uhura and company are FRANTICALLY paging through old Klingon glossaries, manuals and dictionaries.]
UHURA: (subtitled KLINGON) We art delivering food... things and...supplies to Rura Penthe... over...
KLINGON VOICE FILTERED: (subtitled KLINGON) Don't catch any bugs!
[The VOICE LAUGHS RAUCOUSLY. Spock gestures. Uhura returns the laugh...over and out. They look at each other.]
SPOCK: Was that so bad?