Your solution of drilling a hole in the cement floor and installing a drain is a possibility but the normal disposal of the waste water from a softener is through the main house drain lines.
Water softener no drain in garage.
If you choose that option be sure to elevate the softener so you can drain it and still have space to install a sweep into the stack pipe.
The drain will need to be higher than the wye.
According to most plumbing codes a water softener drain line must not be connected directly to the household waste system.
By code water softeners must be emptied into a laundry sink floor drain sump hole or properly trapped special outlet preserving an air gap of at least two times the diameter of the drain line but in no case less than 1 1 2 above the top of the.
Only other alternative i think for a drain outlet would be to run a pvc line along the garage line from the softener drill a hole in the wall brick fascia and take outside on to the grass or into a sometime of drain on the lawn would have to check the local code.
Lemonplumber fl no gate valve on the inside wall or across the garage near the water heater 1 2 is too small a pipe and would not be used to supply the whole house sounds like you will need to do a little digging to verify what and where the feed line is good luck dig gently most pvc services do not react well to a shovel.
If it s any help the brine tank can be located up to 20 feet away from the softener.
A softener drain line must not be connected directly to the waste system but must be emptied into a laundry tray floor drain or properly trapped special outlet preserving an air gap of at least two times the diameter of the drain line but in no case less than 1 1 2 above the top of the receptacle used.
My guess is may be wrong both the washer hose and the water softener drain lines were pushed down the 2 opening.
Water softener in the garage help.
This is to prevent non potable water from flowing backwards possibly mixing and contaminating potable drinking water.