Note: I do not know whether this is the best section to post this topic in.

So quite a simple question, could I create particle effects with Maya or 3DS Max like with FumeFX or something and then import in my game?

Most game engine middleware come with tools for making particle effects, so you would use these instead of external software.

- http://unity3d.com/unity/quality/special-effects
- https://docs.unrealengine.com/latest/INT/Engine/Rendering/ParticleSystems/index.html

If you're building your own engine and you really want to use an Autodesk software for particles, then you can export data from those programs by writing scripted utilities for them with MEL (Maya) or MaxScript (3ds Max), though there's much more depth in developing your own particle system and making a visual editor for creating effects with it - it allows you more control and customization as to the effects that you can achieve, and integration with your custom game engine would be very good.

FumeFX is a particle plugin with specialised rendering for use in Maya and 3ds Max - you can't transfer the "look" of those particles to your game, since they are intended for photorealistic rendering at non-interactive speeds.

I am aware that game engines come with their own particle system, but there is a lot more you can do in other programs with particles then you can within the game engines, and that is the main reason why I wanted to know if I could use them.

There are a lot of different game engines and content creation software packages.
If you can please be more specific as to what engine and software that you're considering to use then we can answer more objectively.

Game engine particle systems are something that work real time and look good from any angle.

3DS max / Maya can render particle animations over the weekend to a chosen 2D format and while it's no surprise they could often look better it's also likely that they aren't usable in your game environment in the same way. Sure if you render it in plain 2D frames you can project them onto a plane and cycle through in a game engine. But that volumetric smoke effect will look flat when you treat it this way and camera moves around in relation to the plane.

It's better to for example to render couple puffs of smoke and let the in-game particle manager create, move, rotate and adjust those particles around. I'd strongly advice to use any built-in features and established pipelines to create particles for your game engine.