Before I get started with Quest I want to know if the game can be published on my own website instead of text adventures.co.uk. I can't seem to find that answer. Does anyone know?

I don't think there is any control as to where you post / publish your game / program...
(It is your work after all!)
But, by publishing your work here, you get feedback for people that work with Quest...
Because they will need to download the Quest program to run it anyway...
(unless it is for a narrow market, IE: fetish, there is no reason NOT to put it here.. just make sure
it is marked as adult, and any other warning as needed...)

As DarkLizerd said, you may post your creations wherever you like, however if you mean publish as in for online play the way you can play games directly on the quest website, then the answer is "no" with a "technically yes".

What I mean by that is that you CAN actually host the games online on your own site, but it is rather difficult and I am not aware of anyone getting it running successfully. You need to run your own server and setup the actual webclient code (or whatever the correct term here is) to run properly on your server. Which is as I said before, rather difficult

To phrase it another way, you're allowed to publish your games anywhere: Quest is open source, no strings attached. But setting up a server so that people can play Quest games from your own website runs into the technical issues described by TinFoilMkIV. I recommend doing the next best thing and embedding the game from here into an iframe. That's a literal one-liner in HTML, and most visitors won't be able to tell the difference. Hope this helps!

You won't be able to publish your game as an online game anywhere other than textadventures presently; although that should change with later versions of Quest.

Here's the technical know-how required, due to how Alex created/coded quest and its online needs/scope:

(probably why 'Manowar' is having server issues causing the issues with online play, and 'Pixie' is trying to learn what he can of this stuff, so he can get more involved with quest development/advancement)

https://blog.textadventures.co.uk/2016/12/07/looking-for-a-new-owner-for-textadventures-co-uk-and-quest/
https://blog.textadventures.co.uk/2016/12/12/developers-needed/

If doopdoopdoop just wants a separate place to display his games, then he would not need the server set-up...
He would just need to tell his players where to get Quest to run the games...
I don't think (I would hope not) he wants to "host" them so that they ran on-line...
Just download and play...
(could he provide a downloadable copy from his site without any problems? or should he provide a link to this site to download Quest?)