I expect its the captain's prerogative to dictate his logs in whatever tense he deems appropriate, just like writers do. Usually thought, writers write in *past tense*. Additionally though it isn't shown, it is highly probable that as Picard dictates his log he is lining up his log with sensor readings from the computer, and as such the present tense seems more smooth than past tense. He is (re)"viewing" the Enterprise being halted by Q, and dictates that "Our mission...has been halted...".


Out of universe this adds suspense to the viewer as present tense gives us the sense that we don't know what is happening next.