The relevant scene described by the OP takes place at the Veridian system. According to the Star Trek: Star Charts (p. 65):
Veridian was classified as a G-class star.
All G-type stars have a mass of 0.84 to 1.15 solar masses and surface temperature of between 5,300 and 6,000 K, so we can safely assume that Veridian is roughly similar in mass and temperature to our Sun. Therefore, any habitable planets will be at a distance of roughly one astronomical unit (AU) or so, which corresponds to the "Goldilocks zone".
Then, assuming the distance from the planet to its star to be around 1AU, the probe should be travelling at warp ~3 in order to hit the star in around 10 secs.
The reason for this is that 1AU is ~8 light minutes or ~480 light seconds, so for the probe to cover the distance in say 10 secs it has to be travelling at a speed of 48c (c=speed of light). According to the wiki and the warp diagram from the TNG Technical Manual seen below, 48c is about warp 3.
Since Starfleet's Class 8 probes can reach warp 9 and are no bigger than a photon torpedo, it is certainly plausible for a small probe like the one used by Soren to reach warp 3 given the available tech at the time.