Is carriers seeking out enemy mining stations an automated behaviour?
Sort of, and not really specific to carriers. Warships of any type will almost always go after hostile targets in the same system (mining stations, research stations, resort bases, defensive bases, spaceports, enemy warships, freighters, etc) if the warships are idling. If the ships are automated, there's also a chance that the computer will decide to dispatch them to attack "appropriate" targets elsewhere, if the computer knows that the player knows about the target (e.g. a station that one of your ships has detected, say by being in the same system or passing close enough to detect it with proximity sensors if you equip those, or a fleet within the detection range of one of your long-range scanners; you the player may "know" of additional potential targets because you've seen ships belonging to your opponent moving towards it, but unless you've actually detected the target before the computer will assign ships or fleets to attack the target). Automated pirate warships with assault pods can be dispatched to raid colonies and bases, and any warship with assault pods might decide to try to capture an opponent if the ship automatically engaged the target rather than being ordered to attack.
However, I will warn you that the computer's target prioritization and definition of appropriate can be questionable. You can see fleets of heavy ships assigned to strike little mining stations out in the middle of nowhere, for instance, and you can also see lone small escorts assigned to hit freighters (which is fine) that are sitting next to size-6000 large spaceports (which is somewhat less fine). Unless you have automated fleet formation active, the computer will never dispatch individual ships of a fleet against targets outside the system in which the ships are present (if you do have automated fleet formation active, then the computer can and will change the composition of your automated fleets and might, as a result, end up with additional unfleeted ships or more smaller fleets than you gave to it, which can result in the fleets you created being split up, with the parts being sent on different missions). Putting your automated ships into fleets and assigning postures can help combat these issues (though note that postures do not work with automated fleet formation active, as turning on automated fleet formation allows the computer to change the postures of automated fleets), as postures and the associated range settings can allow you to control where the fleets will operate, and grouping the ships into fleets reduces the likelihood that the computer will assign individual ships to missions where they're hopelessly outclassed (granted, this cure can sometimes be worse than the illness, if it decides that your fleet of 10 size-230 escorts was an appropriate force to go hammer that freighter that's sitting next to a size-6000 large spaceport, and of course this also means that the computer is more likely to assign an inappropriately-strong force to a mission since it has fewer weak units available for assignment). If you set an attack posture on a fleet, remember to set the attack target to something that the fleet cannot destroy (which usually means a colony), because if the fleet does destroy the attack target the game no longer knows where the center of the area covered by the attack posture should be, which results in the fleet behaving like any other automated postureless fleet (i.e. the computer can send it wherever it wants to send the ship). This also applies to a lesser extent to fleets in defensive posture; if the home base is lost, the game no longer knows where the region covered should be centered and the fleet becomes a free-roaming automated fleet, though as home bases are likely to be colonies (which are also the only home bases which can be assigned through the fleets menu) or powerful space stations (because it's often the case that even though those mining stations are nice to have, they're not really worth a dedicated guard fleet), there's a decent chance that a fleet which has lost its home base has also lost a reasonable fraction of its ships and is engaging the enemy in the area where its home base used to be.
their firepower is slowly being whittled down with piecemeal attempts to break my stuff.
You may want to keep an eye on that in case the Quameno send something actually threatening; that would be a good opportunity to go kill some of their spaceports while their fleet is busy killing one of your stations, or if you feel you can manage it you might try breaking the Quameno fleet (though remember that you will need to be able to capitalize on the destruction of their fleet by killing their spaceports shortly afterwards, or they will just rebuild and you will probably have taken reasonably heavy losses for no particularly great long-term gains).