No. Atlassian describes Sourcetree as a free Git client for Windows and Mac only, and offers no Linux download. You will need a different Git app on Linux.
Sourcetree by Atlassian · Coding · No way to run it
Sourcetree's own page calls it a free Git client for Windows and Mac, and the only two downloads on it are
a Windows version and a Mac version. Atlassian has never shipped a Linux build, so there is nothing to
install.
This costs you less than it sounds. Sourcetree is a front end for Git, and Git itself is the same program
on every system. Your repositories, branches and history live in the repository folder, not in Sourcetree,
so a different app opens exactly the same work with nothing lost or converted.
GitKraken is the closest replacement if you want the same kind of graphical history view, and it does have
a proper Linux version. If you spend most of your day in an editor anyway, the version control panel built
into Visual Studio Code covers staging, committing, branching and pushing without a second app.
If you already use a JetBrains IDE, its built-in version control window does the same job. Toolbox is how you install and update those IDEs on Linux.
Your files: Works on your existing repositories unchanged.
This doesn't have to stop you
One stubborn program doesn't have to decide the whole question. You can keep Windows on the same PC and restart into it only when you need Sourcetree. Everything else you do can move.