So I’m not entirely sure where to put the ground wire. Do I need to buy a ground screw for that hole? Or is there something else I can attach it to? 
4 prong wire outlet, 3 wire dryer – can’t find ground screw
dryergroundwire
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DO NOT connect the ground wire to the grounded (neutral) conductor, as this could lead to current flowing through the body of the dryer (and potentially through you).
The installation guide for the dryer will have wiring instructions for both 3, and 4 wire configurations. Check the manufacturers documentation for proper wiring, but I would say the first image is likely correct.
Instructions from random Maytag Installation Instructions (PDF)
3 Wire Cord
4 Wire Cord
Update:
After doing some research, and looking at dryer wiring diagrams. It turns out that the green/yellow wire is not a ground wire, it is a neutral to case bonding wire. When this wire is not in use (in a 4-wire installation for example), it is simply connected to the neutral terminal and is unused.
That would be fine, as long as you make sure that unused ground wire is in tact and properly terminated back at the panel.
Testing continuity to ground is not IMO a sufficient test, since a neutral and a ground would both show continuity to ground. Hopefully the cable is run exposed where you can see it and verify that it goes where it should in the panel.
Related Topic
- White wire when changing from 4 prong to 3 on dryer
- How to reattach the “head” of a ground wire
- Grounding a 3-prong dryer plug
- Electrical – connect the ground wire for the dryer to a different hole in the chassis
- Electrical Dryer Wiring – Fixing Short Ground Wire in 4-Prong Dryer Cord
- Electrical Wiring – Ground Screw Inverted: Converting 3-Prong Dryer to 4-Prong
Best Answer
The dryer's instruction book will have instructions on how to change the cord. The most important thing is that right now (with 3-prong), there is a "ground strap" connecting neutral and ground on the dryer. This must be removed.
I think I see it right there; a metal foil that lands on the neutral terminal and connects to chassis. You must remove that. Feel free to use that screw for the ground.
If you have a volt-ohm meter, you can sanity-check this by measuring resistance between neutral and ground pins on the recep. It should be high resistance (0L or infinity ohms means high resistance).
It sounds like you're already onboard with this, but you must change the cord on the dryer to a 4-prong cord. It is illegal to continue with a 3-prong connection or to defeat ground in some way.
Old 3-prong connections are dangerous. If there is any trouble with the neutral wire, the dryer chassis will be energized with voltage. (meanwhile the washer right next to it will still be grounded, so touch both and whammo!)