It is extremely unlikely that the D-Link device will be able to wirelessly extend the signal from your AirPort router.......since Apple uses proprietary settings that are designed to only allow other Apple devices to extend or repeat the network.
I have not tried the specific D-Link model that you have, but other D-Link models that I have tried were not compatible with the extend feature on the Apple routers. Of course, anything is possible though.
The AirPorts do not have a WPS button, but they do have a "First Press" option that works as substitute for a "button". I'm going to assume that you are not still using the 5+ year old OS X (10.6.8) that you show in your profile, and that you have updated to a more recent operating system.
Open Finder > Applications > Utilities > AirPort Utility
Click on the picture of the AirPort router, then click Edit in the smaller window that appears
Click the Base Station menu at the top of the screen
Click Add WPS Printer
Click on the image below to enlarge it.

Whether or not that will allow the D-Link to work with the Apple router is open to question. You will know after a few tries.
If the D-Link will not extend the network......as I suspect....it would be best to return the device and use another Apple router......like an AirPort Express.....which you can set up very quickly and easily in a few minutes.