I have not seen such support integrated into gqrx.
I would suggest you investigate the use of a bash script and 'xdotool'.
The following thread:
gives some hints for starting gqrx after the UI starts and then using netcat to send the command to start DSP.
Then some googling gives hints on how to maximize the window:
The 'AI Overview' I got was actually pretty close to what I would do.
So, it will take a bit of linux shell and UI scripting, but I think it's doable.
Oh, and if your distro is using Wayland instead of X, I don't know how to do it other than switching back to X.