It is true that it is possible to contract an STI through a blood transfusion. This is why you should avoid using syringes that other people (especially if it's somebody you don't know) has been using because you can contract many diseases from them, not just an STI. Some of the examples of such STI would definitely be HIV, so if you don't want to go down with it, always have that blood transfusion checked.
The answer to your true or false is that this statement is true