Sure. But what's so odd about that? Star Trek spacecraft are really fast when in space, but tend to move very slowly within atmospheres; this probe need not be an exception at all. Earlier in the same movie, another probe had been launched towards another sun, this time from a space station, and had apparently completed the journey in seconds as well. Warp technology should allow for that.
...Even if for some reason even very high warp factors appear to amount to a crawling speed in the proximity of stars. Say, our heroes go warp eight or even warp ten in their Klingon BoP in ST4, yet it takes the ship several seconds to arch around the Sun nevertheless. But that's not a major problem as such, because probes can be assumed to be faster than ships. In many cases, they indeed appear to be.
The science goof there is what happens after
the probe reaches the sun. The sun dies - and Soran and Picard immediately see it happen! Light from the star (or darkness from the star) does not travel at warp speed - it only travels at the speed of light. So it should take five to ten minutes for the death scene to reach Veridian III, not a split second.
(Of course, we can argue that there was a cut in the action there. Two old geezers had just fought each other, and one had triumphed and killed a star. What would there be for the two to do
except catch their breaths and wait for the death scene to become visible? Picard would gain nothing by climbing the cliff to give Soran one more punch in the jaw; Soran would gain nothing by coming down. Perhaps the two shouted insults at each other in hoarse, huffing voices for five minutes, and the director mercifully spared us from that.)