The FIX lies not within Brave, but you need to modify the standard macOS Keyboard / Shortcuts / Keyboard setting for “Move focus to next window”.
- open macOS System Preferences
- click on the Keyboard panel
- click on the Shortcuts tab within the Keyboard panel
- click on the Keyboard setting on the left within the Shortcuts tab
- now change the Move focus to next window shortcut by clicking on it first, before clicking into the pre-set shortcut on the right and then performing the desired command for switching or toggling between open windows.
-For me it was always cmd < for moving the focus to the next window
-And cmd shift > to toggle or switch back. (This doesn’t need to be additionally set, it just works as well)
When setting up a new macOS (at least in older macOS versions), this setting is at default. (Not sure about the newest releases.)
The cmd ’ was actually even pre-set as default, but didn’t work either for me in Brave and seems more cumbersome to use and also even more cumbersome to setup the switching back feature. Also cmd shift ’ opens a new tab with the Brave Help Center.
Last litte tip: check out the other settings in that Keyboard menu… ctr fn F4 toggles ALL available windows from all open apps etc. or moves focus to the dock or other input or menu areas 
Be Brave