To go on "all fours" is a method of walking that animals do. A grizzly bear can stand up on two legs to look at something, but then go down on all fours to walk. To get down on one's hands and knees is the expression when someone is kneeling with their hands on the ground (for example if they were cleaning a floor).

The correct expression is to go or be 'on all fours'. Note that there is an 's' at the end of the word - it is FOURS, not four. It means to be, or to move, on your hands and knees. Crawling, in other words. This is the only meaning of this expression. I also found the very strange definition you mention about 'in conformity with', but it is only on Dictionary.com - no other dictionary has this definition. It's very odd indeed. I can only assume that it's some kind of mistake. Forget about it.

I have no idea where you found the definition related to "conformity", because I can't imagine how it could be used in that way. The expression is "to go on all fours", with the "s" at the end of "four". It means as you say to "go" or "move" on hands and knees.

Thank you very much for your all the help. Now I'm clear about the meaning of this phrase.