My wife bought some LED Strips that we use to accent the living room and underneath kitchen cabinets.
Problem is, the ones she bought have a bluetooth controller that I presume isn’t compatible with HomeAssistant.
These are 4 prong connecting LED Strips. Are there any LED Controllers that work with HomeAssistant so I can get these strips to work with HA?
If they are addressable LEDs you wont find better than WLED.
It does do RGB lLEDs - with some extra hardware.
How interested in DIY are you? 
If you are (and you should be 
), ESPHome is the perfect solution for you. You connect the LED strip to your ESP8266 or ESP32 and can flash it with your own firmware (ESPHome).
AS much as I love ESPHome, for a LED controller WLED is better.
I found this:
https://www.hadevices.com/product/124
Pretty handy website, actually. Has a lot of recommendations for various devices that work with HA.
Totally agree, but if someone wants to start into the “ESPbased universe”, I find it less scary with ESPHome. 
It integrates better with HA and the installation and flashing procedures are easier, at least if you run HomeAssistantOS or supervised.
And, when not so common with DIY, here an another idea.
I use this in my hosue a lot.
How does it integrate into HA? is there an integration or it is a YAML configuration
It works with the LimitlessLED integration.