Another option is to laboriously fill in the dent with spackle, layer by layer, and then carve in a matching 'grain' with a paperclip or other such tool in the topmost layer. Then repaint, and it'll look fine.
This would take time and patience (each 1/2" layer needs to dry for a while), and be an excellent learning experience for SOMEONE.
No that quote is too high, especially the 300 side of the quote. You are paying a company, not the installer who gets less than half of that. You should seek out an installer and be able to get that done for ~100, which is reasonable.
Doing the job yourself will be a little more difficult than you might want to get involved in.
First you need to get the knee kicker, then tack strip, some reinforcement nails, something for the room transition (metal, wood, etc.).
Second you are dealing with Berber carpet, which will be difficult to stretch and cut. The berber carpet is stiffer than regular cut pile carpet, so any bubbles/wrinkles will be harder to kick out. The continuous fiber strand will be prone to snagging if not fully cut, pulling across the room and leaving a gap. Plus the chance of leaving cut marks on newly finished baseboards.
But this can be done if competent enough.
When cutting the tack strip, you can use tin snips. It won't damage them for the few cuts you will make. Lay the tack strip against the wall with a distance of 1/2" (approximately the width of your finger). Make sure the tacks are leaning towards the wall, so as to catch the backing of the carpet when stretched over it. The gap you left is where the carpet will be tuck into when trimmed.
If nailing into concrete, make sure you use smooth deliberate hammer strokes so as not to have nail deflection. If nailing into anything harder than concrete, use a 5/16" masonary drill bit and use 3/4"stub nails to secure the tack strip down. Use reinforcement nails where ever necessary.
Before laying the carpet down, make sure the padding is up to but not on the tack strip. Trim if necessary.
When stretching the carpet with the knee kicker, try not to drag your knee or you'll end up with a really sore knee/carpet burn. After each knee kick you will rub/press the carpet onto the tack strip, using a hammer that is clean, to secure it. Keeping the knee kicker perpendicular to the wall, when kicking.
For trimming the carpet, you will want to have a very sharp knife and take your time. For this small of a job you can use a utility knife. Fold the carpet back from the wall without pulling the carpet off the tack strip. Without pressing down too hard on your knife, make a cut a 1/2“ from the fold along the wall. If you have trouble cutting the Berber strands, you can cut them with a scissor. When done with a section, tuck it in with a spackling/paint tool (a 3-n-1 will work as well).
Just take your time if you do this on your own.
Best Answer
You can repair it but you're not going to be able to match the wood decor grain - which means you'll have to paint it.
There are a few different products you can use but I like Bondo. It's basically the same stuff they use for auto body repair. It looks like there is enough backing in the hole that you shouldn't need anything but the Bondo, a putty knife and a few sheets of sandpaper. Apply it according to directions trying not to overfill it too much since you'll want to sand it smooth with the door. After it dries start with about a 120 grit sandpaper or sanding block to take off most of the excess and finish with a 220 or higher grit to finish. If there are any hollows in the patch you can apply more Bondo after it's dry. It will be difficult to get the patch to totally disappear but with patience and going over it a few times you can get it very close.
It's hard to tell the material in the picture but you may need a good prime coat to get the adhesion and consistency you'll want with the paint.