Hi there, thank you for your question.
No, pus on the tonsils does not necessarily indicate a sexually transmitted infection (STI). Chlamydia and gonorrhea of the throat are actually most often asymptomatic.
Pus on the tonsils is usually caused by commonly passed viruses. If you are concerned, you could visit your family doctor for an assessment. And if you have an elevated chance of getting an STI in the throat (ie. you are a person who gives oral sex to a person with a penis) then you could ask your doctor to test for chlamydia and gonorrhea of the throat to rule those out.
Let us know if this does not answer your question or if you have any more questions or concerns.