Most lower body exercises that you do in the gym are performed with two legs, such as squats and deadlifts. These exercises don't challenge your balance very much.
I'm not even going to talk about leg extensions and leg curls on machines because you should not be doing these useless, harmful exercises. Especially avoid the leg extension machine since as a mixed-martial artist, you've probably already got some minor knee issues from grappling, so you want to avoid anything that stresses your knee ligaments, such as the leg extension machine.
So if you want to train your single-leg balance, you have many options. Lunges are a good example, but lunges are a dynamic movement and you're only on one-leg for a very short period of time.
Instead, to really work the stabilizers of the hip and ankle, you want to do exercises that have you on one leg while other body parts are moving, causing you to really develop your balance. 3 great exercises for doing just this are the 1-arm row on 1-leg, 1-leg stiff-leg deadlift, and 1-arm overhead press on 1-leg.
All 3 exercises are just like their 2-leg counterparts. You can find out how to do these exercises with a simple search.
The benefit of adding these exercises into your routine are that you'll develop the stabilizers around your ankle and hip, as well as your core.
But don't replace all of your standard exercises like squats, deadlifts, and presses with these single-leg versions, because to develop strength, the standard versions are better choices.
Simply add these exercises in after the big compound movements as a supplementary training method to improve balance and help prevent injury.
Avoid the 5 most common training mistakes that fighters make and learn about a complete MMA workout program that will give you total confidence in your strength and conditioning.