Short answer: Yes and no.
"Al-Qa'im" means "the one who rises up" [for instance, against injustice]. It is primarily used for the Mahdi, but it can also be used for all the Imams.
The literal meaning of the word "al-Qa'im" does not specifically relate to one's name being unused; it comes from the word meaning "to rise" or "to stand".
However, titles for the Mahdi (including al-Mahdi and al-Qa'im) have been used instead of his personal name (which is the same as that of the Prophet (S)) because there are narrations saying that you should not refer to the Mahdi by name. So, in that sense, the Qa'im will arise when people are not referring to him by name; rather, they are referring to him as the Mahdi or the Qa'im.
[For instance, see https://www.al-islam.org/kamaaluddin-wa-tamaamun-nima-vol-2-shaykh-saduq... ]
It is also mentioned in a couple narrations that the Mahdi is called "al-Qa'im" because his occultation will last for a very long time. Therefore, when he reappears, people will have forgotten him so much that it will be as if he is rising up as a new creation. [Similar to how, on the qiyamah or Resurrection, people will rise up anew after being dead.]
[For instance, Bihar al-Anwar, vol. 51, p. 30 - سمي القائم عليه السلام قائما لأنه يقوم بعد موته ذكره. - "he is called the Qa'im because he will arise after his remembrance has perished".]
This is similar to narrations which say that the Mahdi will reappear in a time when faith is at an all-time low. So, in this sense, he may not be commonly spoken about before his reappearance.
So, you can say that both:
(a) The Mahdi is called "al-Qa'im" because he will lead an uprising [this is explained, for instance, by Imam al-Sadiq], and
(b) The Mahdi is called "al-Qa'im" beacuse he will arise at a time when his name has been forgotten, and therefore will be rising up anew.
The first view (that he is called al-Qa'im because he will lead an uprising) is the more commonly understood meaning, but both views are supported by narrations, and one can say that he is called al-Qai'm for both reasons.
Hope that helps!