"My grandmother tries to become a famous designer."
Translation:Bà của tôi cố gắng trở thành một nhà thiết kế nổi tiếng.
No you can't. I grew up speaking Vietnamese and can't really explain how I know. If you did use thử, it sounds like you wanna try something "casually" or "for fun". Or even trying to "test something out". Like "try my cookies", "yeah I'll try that", or "I'll give it a try". Cố gắng to me has a "greater effort" kinda tone. A more serious tone. "He tries to get get his Ph.D", "she is trying to save him", "I will try to do my best". I hope that helps. I wish a native speaker could explain in greater detail as to why we would choose one over the other.
Not "thử" but "ráng" is acceptable. Also "của" translated from "of" is optional and often omitted.