You are right, Google Translate is wrong.
“Ne pas avoir à se plaindre” is quite idiomatic and means approximately “to have no reason to complain”, i.e. to be satisfied with a general situation. The example from Alexandre is a great use of it.
- Hows your new job? - Ah! Coworkers are nice, salary is decent, duties are challenging..! Je n'ai pas à me plaindre!
“Not to complain” would be “ne pas se plaindre”, simply.
Moreover, in that case, Google translate misidentified the tense: it translated a past tense into a present tense. You should indeed use a past tense in French too, the passé composé being the most logical here.
In the end, “ne pas se plaindre”, passé composé first person singular is indeed “Je ne me suis pas plaint”.