How can I permanently remove the boot option "quiet splash"?

Open a terminal and type the following:

sudo  gedit /etc/default/grub

Then look for a line like this in the opened file:

GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"

It may have other options there as well, but just remove quiet splash and:

sudo update-grub

Open a terminal (Ctrl+Alt+T) and issue the following command

pkexec gedit /etc/default/grub 

or

sudo -H gedit /etc/default/grub

I assume here that gedit is installed on your system, if you prefer another editor use the appropriate name (e.g. leafpad, kate...etc).

Search, find and edit the line

GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" 

to

GRUB_CMDLINE_LINUX_DEFAULT=""

proofread, save the document and then run in terminal

sudo update-grub 

It's in /etc/default/grub. Make your changes there and update grub with sudo update-grub