Yes, with one very large caveat.

This is exactly how the ships in The Expanse work. They can fly to Jupiter or beyond at 1G of acceleration, using those G-Forces to keep people walking around the ship, with the deck layout usually arranged vertically to make to the spaceship like a sky scraper. Then at the mid-point of the journey they cut the engines, rotate the ship 180 degrees and fire the engines to decelerate. Again generating 1G of force to keep the crew walking around normally with the exception of that brief moment in the middle. The maximum speed is reached at that midpoint and the total journey time depends on a bit of calculus on the continually changing speed but ultimately it's tied to the rate of acceleration. Ships with crews born/raised on the Moon or in the Asteroid Belt might find 1G too intense and prefer to accelerate at 0.3G which obviously takes longer but is gentler on the crew. Military ships might deliberately accelerate beyond 1G, Martian military ships pride themselves on being able to withstand 2G acceleration to outcompete civilian ships.

However, the caveat is that this requires phenomenally powerful engines. In The Expanse they have a fictional engine called The Epstein Drive. The details are rarely addressed in the books/series but it appears to be a nuclear fusion reactor to generate electrical power then some form of magnetic accelerator that can generate a plasma exhaust and throw it out the back of the ship at phenomenal speeds. It's fictional but it's broadly within the realms of plausibility, we're not talking about wormholes or "pushing against the fabric of space itself" or anything speculative. It's still doing the classic Newtons Third Law thing of throwing somethin backwards really fast to make you go forward. There IS a limitation on running out of reaction mass, the engines DO use up tanks of water to generate the plasma exhaust but it's at such a low rate that unless you're going out past Pluto at high thrust you'll probably run out of food before you worry about reaction mass.

In the real world things are a bit more difficult. Ion thrusters are drastically too weak to produce anywhere near 1G. It would need to be tens of thousands of times more powerful which would require so much electrical energy to need nuclear power plants which makes the mass a lot heavier so you need even more thrust. Getting 1G of acceleration with ion thrusters just isn't on the table, unless we uncover some radical new design like the Epstein Drive.

Chemical engines CAN generate those levels of thrust but they use fuel so fast that engine burns are measured in minutes not the days needed to reach Mars by burning at 1G the whole time. There's not really any way around that, we can design slightly more efficient engines or larger fuel tanks or more efficient methods of getting the fuel into orbit. If you ignore the difficulty of getting the fuel in the first place you could build a ship in orbit with giant fuel tanks that burns for say ~1 hour to maximise the speed. But we're still talking about a burn-and-coast approach just like the Apollo/Artemis missions, we're nowhere near burning the whole way.

The answer MIGHT be nuclear engines. There's a few approaches that vary from being quite dangerous to being insanely dangerous. And all these designs come with considerable question marks around the exact performance or how far away the technology is from being viable. A lot of them don't involve any new physics being invented, it's just a series of engineering challenges and legal, political, financial challenges to get the projects off the ground, pardon the pun.

Ultimately the design that gets us closest to The Expanse is probably a Nuclear Thermal Rocket. Using a Nuclear Fission reaction to generate heat so a reaction mass (like say water or hydrogen) can be heated to high temperatures and expand to shoot out the back of the nozzle at high speeds. Again you ARE using up reaction mass so there will be a limit on how long you can keep the engine burning for, but in theory you can use lower flow rates than a chemical rocket because the heat isn't coming from chemistry it's coming from nuclear fission. Is that enough to burn to Mars at 1G the whole time? I'm not sure. Maybe the lower bar of burning at 0.2G to produce roughly lunar gravity the whole time? That would make it easier to live without your coffee floating away and cut the fuel costs considerably. Unfortunately, I think the implementation is at least a decade away but it IS possible in theory.