I've always assumed that Trelane was not related to the Q continuum but it's hard to ignore their similarities (e.g. super-human powers, mischievous personality, testing/observing humans, etc).
Are they related/associated at all?
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Trelane only appears in The Squire of Gothos, in the original Star Trek. He never appears again or is referenced again in canon. (In Trek, canon generally refers to what we see on the TV shows and movies and excludes printed material, but it can vary depending on whom you ask and how rabid they are.)
However, Trelane appears in non-canon works, as described in the Apocrypha section of the entry that covers him at Memory Alpha. He shows up in the game Star Trek: Judgment Rites, and now has an interest in World War I. There's no real indication of his origin in this.
In Peter David's novel Q-Squared, Trelane is said to be Q and is under the watch or mentorship of the Q that we have seen as played by John DeLancie. While I can give you this from Memory Alpha, I can't tell you why this doesn't contradict the issues that arose in Voyager that resulted in the birth of a new Q, which was something new to all the Q.
The Star Trek Encyclopedia, considered a canon source of information about the Star Trek universe ("officially licensed and endorsed by Paramount") directly cross-references Trelane's article to the article about the Q.
Tellingly, the only other articles that have this same cross-reference are known to be members of the Q-Continuum such as Q2, Female Q, Q Colonel and Quinn.
Trelane : Life-form of unknown origin and extraordinary powers. Trelane kidnapped several Enterprise crew members in 2267. A tall, dashing humanoid male in appearance, Trelane was actually a small child from a civilization of noncorporeal life-forms. With his ability to change matter to energy at will, he created the planet Gothos and manufactured an elaborate facade of a Gothic castle from Earth. Trelane patterned himself after an 18th-century Earth squire. He toyed with the Enterprise personnel, eventually forcing his parents to keep him from making any more planets until he could learn not to be cruel to inferior life-forms. SEE: Q
Also demonstrated similarities with trelanes obsession with military titles and the trial he put Kirk on it tos. He also said in that trial that anything Kirk said was already entered in record to be used against him. Implying he had foreknowledge of the outcome, to a point made obvious by the appearance of his parents. Who I said earlier were prob Q and his Q wife.
I don't claim to be right. Most of you probably know more about star trek than I do. The answer I came up with came from the 2 books The Fifty Year Mission: the First 25 years and the 2nd book The Next 25 Years. From what I read several writers from the original series were also used on the Next Generation. Q was basically I rehashed version of Trelane. Though not related or the same kind of being it's just a similar character because coming right out of the gate they didn't have a completely fresh idea.
I don't think Trelane is a "Q". Trelane was not aware that food or wine had no taste, or that the Earth he knew was from 900 years earlier. "Q" would have known all of this. However, at the end Trelane's parents come and save the day. They treat him like a child so maybe he's an adolescent "Q" with limited powers
Although canon does not specify that Trelane is from the Q Continuum, his being a child of his (unspecified) species does not rule it out. The case of Amanda Rogers in the ST:TNG episode "True Q" shows that, while uncommon, Q can have offspring which are also Q. (And certainly they wouldn't be the first lifeforms to do that simply out of boredom).
Trelane seemed to use technology, but his parents did not seem to need any. So I lean towards him being a Q, but maybe with the technology playing some role not described!
Using the idea that the planet was a play pen for a naughty child, perhaps the technology was like a crib to keep the child confined to some degree! Maybe it was there not to grant him powers, but to limit them.
But just like "Q2" , "Female Q", "Q Colonel" and "Quinn".
Q's son's name is nothing more than "Q Junior" not Trelane.
In the whole scheme of things I'd say it's more likely that Q and the Continuum, were inspired by Trelane's character, but have definitely surpassed him in all things.
But there is the off chance he was adopted by someone in the Continuum.