Does the Technomancer's Supercharge Weapon spell work on a ship's weapon for the purposes of ship-to-ship combat?
No, you would not be able to affect a starship weapon with that spell (or most others).
Assuming you've found some way to Touch your weapon safely while in combat maneuvers (no mean feat)...
From the Starship Combat section,
Starship weapons and regular PC-level weapons work on different scales and aren’t meant to interact with each other. If characters choose [sic] to shoot at a starship with their laser rifles (or cast a spell on it) while it is on the ground, the GM should treat the starship as an object (a particularly massive one, at that). At the GM’s discretion, if starship weapons are ever brought to bear against buildings or people, they deal Hit Point damage equal to 10 × their listed amount of damage. However, starship weapons are never precise enough to target a single individual (or even small group) and can, if the GM decides, be simulated as deadly hazards instead of weapon attacks.
By corollary, if a GM even allowed the spell operate on a Starship weapon (which isn't a weapon as the spell would be concerned), it should increase damage by 1/10 of 4d6.