Total necro post but I thought I'd post a solution for those that couldn't make it work through the EA app. The shortcut method in the Wiki still works, albeit with some changes. I'll detail it below for those who need instruction.
Note: This won't work with the Ultimate edition in Steam.
Create a shortcut on your desktop, Start Menu, or anywhere else to the file daorigins.exe, which is now located in "C:\Program Files\EA Games\Dragon Age\bin_ship"
Right click the shortcut and choose properties. In the Target field add the following parameter at the end of the line: -enabledeveloperconsole. It should look like this:
"C:\Program Files\EA Games\Dragon Age\bin_ship\daorigins.exe" -enabledeveloperconsole
Make sure that there is a single space between the last quotation mark and the hyphen.
Partially quoted with corrections from the Dragon Age Console Wiki located here: https://dragonage.fandom.com/wiki/Console_(Origins)
Lastly, go into C:\Users\*username*\Documents\BioWare\Dragon Age\Settings (Or the equivalent for your computer) and find the keybindings.ini file. Open it with either C++ or Notepad and modify the key assigned to Openconsole from Grave to Tilde. Then make sure to download the following mod: https://www.nexusmods.com/dragonage/mods/1110
Enjoy!