Can I design my survey on another platform but distribute it through Qulatrics?

Curious why you're looking to design outside of Qualtrics? I believe the best solution to your question is simply designing and distributing it all inside the Qualtrics platform.

Because I have previously designed on sawtooth and I know how it works when it comes to designing a discrete choice experiment which does not seem to be straight forward on Qualtrics.

I don't know of any way that involves doing things as they're designed. I think you need to have a link to the survey to send a distribution from Qualtrics. What you could try to do is create a one question survey in Qualtrics where the one question never displays based on impossible display logic and then have the landing page automatically redirect to your other survey. You could also include your actual survey as a second link in the body of the email and make the Qualtrics link not obvious to click on. But both of those are kinda strange solutions. I don't know if there's a clean one.

The first suggestion I can definitely try! Thank you.
The second suggestion I am a little confused about...this is my very first time using Qualtrics. I thought the distribution part is handled by the project manager at Qualtrics?

I don't know what you are able to do vs what a project manager does for you, but I am talking about the email distribution under the Distributions tab of a survey project. If you were to go to that page/tab and compose an email to distribute, Qualtrics will require your link to the survey to be somewhere in the email (as that's the reason for the distribution functionality). I would probably recommend that you exhaust your options for programming the survey in Qualtrics the way you want it or just distributing the non-Qualtrics survey outisde of Qualtrics first though.