Basically, I created this low-poly body mesh that can be subdivided into a higher poly mesh with good topology for animation. However, I designed this knee/elbow topology when I was starting out and I don’t think this is good topology. How do I fix the topology? (Please draw on the provided images)
The question is very vague. What exactly do you want to fix? What’s wrong with it exactly? What result are you after? What problems are you trying to solve and why?
Right now it sounds like you want to start fixing stuff that isn’t broken 
I moreso want to fix the topology so it matches the topology of professionally made models for animation, like for games for example. Right now it does not, and more importantly, the poles on the sides are bad.
So add an extra loop and get rid of the poles.
But… poles are unavoidable. The trick is to put them places that don’t matter.
A lot is said about having “good topology” — but in a very real sense topology doesn’t matter. There are principles that will help your models deform nicely, but it is a trap to spend all your time making a perfect topology.
Those knees look OK to me. The question is— do they deform nicely when textured & rigged?
Even if you just slap on a temporary two or three bone rig to bend the leg and a procedural texture, bend it and see how it looks. Unless something is very obviously wrong (visually) then you can consider it done and ready for posing and animating. Time to focus efforts on proper texturing and a proper rig.
No I mean, I want the topology to look more like the topology seen in professional models, like for games and such
There’s no such thing.
If it works, it works.
EDIT: If you are curious, there are always a couple professional models floating around that you can download and examine. The BF always catalogues theirs. You can still download Sintel, for example. The models for Rumi, Zoey, and Mira in Fortnite are floating around somewhere too*. Etc. You will see that there is no One True Topology™.
Doesn’t matter at all what your topology looks like. Matters if it deforms well. There’s no such thing as one perfect professional knee topology, it depends based on what poses you’re planning on needing with the rig
I too, shall state that what matters is that it deforms well, or well enough. Throw some bones in there, see if it looks like a knee when bending, or not, if it needs adjustment, then adjust.
I’ll also add though, this page has some good explanations and examples: http://wiki.polycount.com/wiki/Limb_Topology note the focus on how each example deforms, and why the triangles work well in their examples.
If you look closely at professional games as you gain more experience as a 3d artist, you’ll spot tons of little rigging, clipping, and even normal map errors that make it through to the final product. So don’t just aim to emulate what others are doing, understand what you are doing and what you want to do, and sometimes be willing to compromise on perfection to hit your polycount targets, or your deadline or whatever.
I get an excellent deformation with your knee.
inside • front • outside
bones
The knee bone is a child of the upper leg and has a Copy Rotation constraint on it targeting the lower leg, local space, influence at 0.5.
automatic weights with minor adjustment
The adjustment was to remove influence from the upper leg on the top of the calf in the back of the lower leg (where the arrow points in the images).
Here’s the .blend
UGP6-knee.00.blend (116.5 KB)
Made with Blender 4.5
So… there is no actual problem to fix then. It’s fine, don’t worry about it. Unless you have a reason. You don’t seem to have one. It’s OK to not do stuff for no reason at all.
Have a look at those professional models you are talking about.
Blender official asset bundle:
It’s just a tube, that’s it.
Random model on one of the 3d model selling platforms:
There is some loop going up along the leg, OK, one could do it like that as well.
A model from a popular game:
Just a tube. 
Relax. It’s all good.
I’m adding “Just a tube” to my list of stims. Thanks!
UPDATE: What I actually want is for the knee topology to match the topology of this game character (right), since it has an actual defined kneecap and has a loop under the kneecap.
The issue is, I don’t know how to modify my current topology so it looks like the topology on the right, and furthermore, in the reference topology, the number of spans on the top of the knee is larger than the number of spans on the bottom of the knee, however in my original topology the number of spans are the same (I want to keep it that way) How do I get the topology of my current solution to match the reference? (I also want all quad topology)
Ref Topology.blend (133.5 KB)
I added a reply to my post to show what I actually wanted, please check it out.
You cannot match it perfectly, unless you replicate the topology as it is. To get a bit closer with lower polys and to get a loop under the knee without getting more faces to the thigh you only have one option: inset way:
Or you need to start making compromises and have more faces above the knee, because every loop must go somewhere:
But, as others already demonstrated, you can get good results with just a tube. Here are the knees of my base character, that I use also for games:
Is there a specific reason why you need it to be all quads? Even though modeling is much easier with quads, because you can easily add loop cuts etc, when it comes to shading and deformation, correctly placed triangles often give much better results than desperately forced quad topology. Even subsurf modifier handles triangles well, because it just breaks every triangle into three quads and continues like nothing ever happened. Ngons are the ones you should avoid with deforming characters and smooth surfaces.