Why wouldn't they?
Laser would likely be the best option as far as signal strength is considered, but radio signals wouldn't be impossible to detect with, say, Arecibo or even better with the long base range interferometres (VLA or VLBA). And if they wanted to listen, they would definitely point them straight at Alpha Centauri all the time it was possible.
Not to mention that GTVA has all it's resources apart from Sol system to build as powerful a transmitter as they want, be it a laser, radio or a message capsule/unmanned ship making a long long series of intra-system jumps towards Sol starting from Alpha Centauri. What's the range of destroyer-class subspace drives within a system? Likely quite a bit considering ships jumping around systems like Capella which are HUGE compared to Sol system due to insanely large central star, so let's go with at least 100 AU.
One light-year is 63,241.1 astronomical units. Distance between Sol and Alpha Centauri is 4.37 ly which equals to 276363,607 astronomical units. That makes approximately 2764 100-AU jumps to reach Sol from Alpha Centauri.
If it takes 15 minutes to recharge the jump drives and five minutes to do the jump itself (it might even be instantaneous but we'll go with 20 minutes per jump), assuming there's enough fuel or energy or whatever and the ship holds together for that close to 3000 jumps, it would take about 55280 minutes to reach Sol. 921 hours. 39 days.
Intersystem drives in the long run can be faster than light on average assuming one jump range is longer than roughly 2 AU. 15 light-minutes (commonly used charge-time for jump drives) is about 1.8 AU. I would safely say the ships in FS2 universe can do inter-system jumps longer than that without nodes. So it's not completely unplausible for GTVA to build a ship capable of reaching Sol - perhaps even in reasonable time frame. It probably all boils down to expense of construction of a ship with such vast energy reserves and reliable subspace drives that this was not attempted - even though it arguably could boost morale in the GTVA...