A handyman can do it. When I lived in Philadelphia, there were many miles of sidewalk and there were guy that specialized in repairing them. Check your local paper for "concrete work" and you may find one that does small jobs like that at reasonable prices.
A wider angle photo would probably help so everyone could see what the whole job is. Something that shows the driveway and the brick etc from a wider angle would make it clear what needs to be fixed and how large of an area would be involved.
I still think that an experienced handyman/woman could do it, but they would have to know enough to make sure the strength of the end product is sufficient to be part of the driveway as you mentioned.
However, just having any experienced concrete/cement contractor look at it will give you an instant answer as to what is need, how much it will cost, etc.
P.S. Since it is outside, any contractor can just drive by at their convenience and take a look at it and give you a written estimate. You don't even need to be there. Just give them the address and you're all set.
Before concrete is poured in, you need a stabilized base. Looks like mud in there now. That needs to come out, take it down to where the ground is solid, packed gravel or whatever. Then fill it up to the required height with suitable material and compact it down. Then the concrete goes in. If you put concrete on top of mud it will sink later.
From the photos so far, I can't really tell exactly what the OP has there.
I see what looks like a small "square" (rectangle) with dirt under it, and another area of just dirt, and some brickwork, and some kind of pipe coming up out of the ground. I have no idea what that pipe is, where it goes, if it is really coming up out of the ground in an area that is actually a part of the driveway, etc.
But, yes, if it is that small area that the OP has in mind filling, and if in fact it is part of the driveway that won't be driven on anyway, I agree that a basic cement patch (not too thick) that can be broken out later if needed would be an okay way to fix this. And, if that is all that is needed, any handyman could easily do this.
Any guesses as to what that pipe is that is coming up out of the hole in the photos?
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