Lets break this down a bit.
So this means that if it accelerates in one direction at about a month, it reaches a significant portion of the speed of light
There are a number of assumptions here. Being able to sustain high-G burns is one thing, but reaching a significant percentage of the speed of light is very definitely another.
The first problem is the tyranny of the rocket equation, and the second is propulsive efficiency.
The first tells us that in order to reach very high speeds without needing implausible amount of fuel and reaction mass, you need a rocket with a very high exhaust velocity, and the second tells us that the efficiency of a rocket travelling below its own exhaust velocity is significantly diminished.
What can we work out from this? Well, if you want to be travelling at relativistic speeds, you're going to need antimatter, and you're going to need a lot of it. Fusion just isn't energy-dense enough, and fusion rockets don't have nearly enough exhaust velocity.
This means that the problem you have is not just one of relativistic projectiles whizzing into your inhabited worlds, but the idea that you would allow anyone to put enough antimatter in one place to crack a hole in the crust of Earth.
Here's the solution to your problem: don't let anyone wander around with hundreds of tonnes of antimatter.
Antimatter production is very hard, requiring large amounts of energy, and the only way to do it economically is using huge solar arrays in the inner solar system. It cannot be produced in the required volumes surreptitiously. That's where security and state controls will be. Literally no-one is going to want to be hanging around next to large quantities of the stuff, because the risks are just too high. Nobody really needs an in-system ship that can reach relativistic speeds, and so no-one will ever be issued with enough antimatter to get them to those speeds.
Hell, with any sensible system, people would get either micro or even nano grams of antimatter or none at all. Now there's no danger of anyone sustaining multiple-G burns for a month, because they'll be out of fuel in a couple of days at most.
a significant portion of the speed of light
Space is big, but we know well enough that it isn't empty. Space in our solar system is so not-empty that you can see its not-emptiness with the naked eye, if conditions are right.
You may be trucking along at several tenths of the speed of light cackling about how much of a hole you're going to leave on Mars, or whatever, and there's a reasonable chance you're going to hit a sand grain and it is going to make a very ugly hole in the front of your ship, which will probably already be partially stripped away by impacts of dust and gas.
It is dangerous to the ship to try and fly around interplanetary space at these speeds. Consider that there may not be any viable protection against this kind of damage, and it might even be fatal. Also consider that any interception technique can really very easily use your own velocity against you. Speaking of which,
sustain high-g burns for long stretches of time... it accelerates in one direction at about a month
Now, if you've followed my guidelines above, no-one will ever be able to travel at relativistic speed within a solar system, because it would be stupid and dangerous and probably obscenely expensive as well.
But what if they tried?
Well, if the aforementioned rocket has an exhaust speed that's a significant chunk of lightspeed (and it needs to) and it is accelerating at several gravities, then you're going to be able to see its exhaust jet from the Oort cloud. It needs an engine power of ~45 megawatts per kilo to manage a single measly G with an exhaust velocity of .3c (the effective exhaust velocity of a beam-core antimatter rocket). A thousand tonne ship would therefore need a thrust power of a little under half a petawatt. But wait! Antimatter rockets of this kind release something like 60% of their annihilation energy as gamma rays, so your total power is going to be more like 1.6PW.
This will be obvious to everyone, everywhere in the solar system with a line of sight. And lightspeed being what it is, it'll be visible long before you get up to relativistic velocities.
And you know what will happen next.
Project Rho mentions the phrase "loudness, lawyers and lasers" and what you could expect for braking space traffic control laws. If you ignore the demands, and the legal threats, then what happens next is that you will be promptly destroyed. There are a whole range of ways to do this, but suffice to say that if you can accelerate your ship to silly speeds then there will be purpose-built relativistic interception systems that will reduce you to a cloud of plasma before you can say "I wonder what's generating all those gamma rays"
To summarise, then:
- You won't be able to get to up dangerous speeds, even if you wanted to.
- Even if you could, everyone will notice well in advance of any collision, and you'll be vaporised.
Now, there are various ways of getting around a solar system promptly that don't necessarily require everyone to drive relativistic kinetic kill vehicles and carry planet-cracking amounts of antimatter. Feel free to ask a separate question about them, because this answer is already too long.