If I run my Selenium tests in Ubuntu + Chrome, will that ensure that my application will work fine on Windows 10 + Chrome?
Chrome is usually operating system agnostic, but there can be differences in the way the OS interacts with the browser, and rendering is definitely subject to different rules.
If your application does not use complex scripting and does not use much if any native functionality (such as file uploads, printing, and so forth), then the application will probably have no issues on Windows 10 + Chrome if it tests out correctly on Ubuntu + Chrome.
Re ensure - how much uncertainty you are willing to handle, and what is the cost of failure (running on ubuntu but failing on Win10)?
If it runs on ubuntu, you have a reasonable chance it will run on Win10. But is is less than 100%. Running tests on two platforms adds additional cost, only you and your manager can decide if additional cost is worth the additional cost, or if it is cheaper to take the (small) risk.
Risk is also dependent on version of Chrome, desired functionality, and other factors.