Yes, you certainly can create a thread or tweet while quoting another tweet. I'm not 100% clear on what you mean by 'quote tweeting' here though.
You have a few options.
If you want to quote another tweet in a tweet, you can just add the URL of the tweet on Twitter to the tweet in Chirr App, and it will show up in the preview. This approach uses characters, though, you can also use the quote retweet option (if you click on the + sign for a tweet on the right-hand preview panel), and it will do the same, except it won't use up any of your characters. Here's a screenshot of what I mean...
You have a few options.
If you want to quote another tweet in a tweet, you can just add the URL of the tweet on Twitter to the tweet in Chirr App, and it will show up in the preview. This approach uses characters, though, you can also use the quote retweet option (if you click on the + sign for a tweet on the right-hand preview panel), and it will do the same, except it won't use up any of your characters. Here's a screenshot of what I mean...
You can also create a tweet or thread as a reply to another tweet. That's different. If that's what you want to do then you can use the 'reply to a tweet' button in the top left corner of the editor. This will open up a window where you add the URL of the tweet you want to reply to. And then your tweet or thread will be posted in response to that message. Here's another screenshot to clarify...
