Firstly, to verify the truth of this statement, we must question what exactly a "fuck" is...
In context, a "fuck" seems to be something that is given. If someone gives a fuck, then we could say that someone effectively gives care towards a situation. So, by not caring about a situation — like someone breathing on you and inadvertently moving your hair, on a day that you would rather not such a thing happen, then, by not caring — one is not giving a fuck, not getting irritated and maybe not throwing coffee in the other person's face. Thus, no stress results from not caring or not giving a fuck as the powerful wisdom of this picture quote would have us believe!
Finally, and as far as the truth of this statement goes, only will time tell if this courageously inspirational picture quote is true or not. For, we need only remember, as the old saying goes:
"Truth is timeless and opinions are timely and the truth that we are capable of not giving a fuck, not feeling stressed is shown within our inability to feel unwell about life, ourselves and others — occasionally. And, we're only that lucky."
—Anonymous